University  of  California  •  Berkeley 

THE  THEODORE  P.  HILL  COLLECTION 

of 
EARLY  AMERICAN  MATHEMATICS  BOOKS 


«    . 


ELEMENTS 


OF   THE 


DIFFERENTIAL  AND    INTEGRAL 


CALCULUS. 


BY    CHARLES    DAVIES,    LL.  D., 

1UTHOR    OF   ARITHMETIC,   ELEMENTARY   ALGEBRA,  ELEMENTARY   GEOMETRY 
ELEMENTS    OF   SURVEYING,  ELEMENTS   OF   DESCRIPTIVE   GEOME- 
TRY,   ELEMENTS   OF  ANALYTICAL   GEOMETRY,     AND 
SHADES       SHADOWS,   AND   PERSPECTIVE. 


IMPROVED  EDITION. 


NEW   YORK: 

PUBLISHED  BY  A.  S.  BARNES  &  CO. 
No.  51    JOHN    STREET. 

1854 


Entered  according  to  the  Act  of  Congress,  in  the  year  one  thousand 
eight  hundred  and  thirty-six,  by  CHARLES  DAVIES,  in  the  Clerk's  Office 
of  the  District  Court  of  the  United  States,  for  the  Southern  District  of  New 
York. 


PREFACE. 


THE  Differential  and  Integral  Calculus  is  justly  con- 
sidered the  most  difficult  branch  of  the  pure  Mathematics. 

The  methods  of  investigation  are,  in  general,  not  as 
obvious  nor  the  connection  between  the  reasoning  and 
the  results  so  clear  and  striking,  as  in  Geometry,  or  in 

the  elementary  branches  of  analysis. 

i 

It  has  been  the  intention,  however,  to  render  the  sub- 
ject as  plain  as  the  nature  of  it  would  admit,  but  still, 
it7  cannot  be  mastered  without  patient  and  severe  study. 

This  work  is  what  its  title  imports,  an  Elementary 
Treatise  on  the  Differential  and  Integral  Calculus.  It 
might  have  been  much  enlarged,  but  being  intended  for 
a  text-book,  it  was  not  thought  best  to  extend  it  beyond 
its  present  limits. 


4  PREFACE. 

The   works   of    Boucharlat    and    Lacroix    have    been 

t. 

freely  used,  although  the  general  method  of  arranging 
the  subjects  is  quite  different  from  that  adopted  by 
either  of  those  distinguished  authors. 

The  present  is  a  corrected,  and  it  is  hoped  an  improved 
edition.  The  first  chapter  has  been  entirely  re-written,  and 
some  of  the  other  parts  of  the  work  have  been  considerably 
altered. 

WEST  POINT,  March,  1843. 


CONTENTS. 


CHAPTER  I. 

n*e. 

Constants  and  variables,      .         .         .         .''*••'.  9 

Functions  defined,  .         .        v        .       *••  "'•'•  *-       »  9 

Increasing  and  decreasing  functions,            '•      .  ,        •  10 

Implicit  and  explicit  functions,         .     '•  .,- d    •  .'        •  11 

Differential  coefficient  defined,             .         .         i         .  16 

Differential  coefficient  independent  of  increment,      .  20 

Differential  Calculus  defined,       .         f         <         .         .  22 

Equal  functions  have  equal  differentials,            w        .  23 

Reverse  not  true,         .         .                  »                  «  ,4>     .  23 

CHAPTER  II. 

Algebraic  functions  defined,            V     ".       ^  ."•'•      i  25 

Differential  of  a  function  composed  of  several  terms,  .  26 

"           "    the  product  of  two  functions,     .•        .  27 

"           "               "            any  number  of  functions,  28 

"           "    a  fraction, 29 

Decreasing  function  and  its  differential  coefficient,        .  30 

Differential  of  any  power  of  a  function,  .         .       •  .    .  30 

"           of  a  radical  of  the  second  degree,      .         .  31 

"           coefficient  of  a  function  of  a  function,    .  33 

Examples  in  the  differentiation  of  algebraic  functions,  34 

Successive  differentials — second  differential  coefficient,  39 

Ta-ylor's  Theorem, 43 

Differential  coefficient  of  the  sum  of  two  variables,  43 

Development  of  the  function  u  =  (a  -f-  a?)",            .         .  46 

second  state  of  any  function,     .  47 

Sign  of  the  limit  of  a  series, 47 


6  CONTENTS. 

Page, 

Cases  to  which  Taylor's  Theorem  does  not  apply,    .  48 

Maclaurin's  Theorem,          .               '    ....  50 

Cases  to  which  Maclaurin's  Theorem  does  not  apply,  53 

Examples  in  the  development  of  algebraic  functions,    .  54 

CHAPTER  III. 

Transcendental  functions — logarithmic  and  circular,     .  55 

Differential  of  the  function  u  =  a*,            ...  55 

"  "  logarithm  of  a  quantity,  .  .58 

Logarithmic  series,          ......  59 

Examples  in  the  differentiation  of  logarithmic  functions,  62 

Differentials  of  complicated  exponential  functions,    .  64 

"             "   circular  functions,        ....  66 

"             "   the  arc  in  terms  of  its  functions,       .  70 

Development  of  the  functions  of  the  arc  in  terms  of  the  arc,  73 

Development  of  the  arc  in  terms  of  its  functions,          .  75 

CHAPTER  IV. 

Partial  differentials  and  partial  differential  coefficients 

defined, 79 

Development  of  any  function  of  two  variables,     .         .  80 
Differential  of  a  function  of  two  or  more  variables,  .  82 
Examples  in  the  differentiation  of  functions  of  two  va- 
riables,   ........  85 

Successive  differentials  of  a  function  of  two  variables,  86 
Differentials  of  implicit  functions,    .         .                  .89 

Differential  equations  of  curves,           ....  93 

Mariner  of  freeing  an  equation  of  constants,     .         .  96 
"                 "       the  terms  of  an  equation  from  ex- 
ponents,          .......  97 

Vanishing  fractions,    .....  .98 

CHAPTER  V. 

Maxima  and  minima  defined,    .....  105 
General  rule  for  maxima  and  minima, .         .         .         .108 

Examples  in  maxima  and  minima,    .         .         ...  109 

Rule  for  finding  second  differential  coefficients,    .         .  112 


CONTENTS.  7 

*  CHAPTER  VI. 

Page. 

Expressions  for  tangents  and  normals,      .         .         .  116 

Equations  of  tangents  and  normals,     .         .  .118 

Asymptotes  of  curves,     ......  122 

Differential  of  an  arc,          .         .         .         .         .  .125 

"           "    the  area  of  a  segment,       ;         .         .  127 

Signification  .of  the  differential  coefficients,          .  .       128 

Singular  points  defined,  .         .         .         .         .         .  132 

Point  of  inflexion,     ;1         .         .         .         .         .  .133 

Discussion  of  the  equation  y  =  b  ±  c(x  —  a)m,           .  134 
Condition  for  maximum  and  minimum  not  given  by  Tay- 
lor's Theorem — Cusp's,    .         .         .         .         .  139 

Multiple  point,    .         .         .         .         .         .         .  .143 

Conjugate  or  isolated  point,      .....  144 

CHAPTER  VII. 

Conditions  which  determine  the  tendency  of  curves  to 

coincide,     .         .         .         .         .         .         .  v  .       147 

Osculatrix  defined,           .         .         .         .         .         .  150 

Osculatrix  of  an  even  order  intersected  by  the  curve,  .       152 

Differential  formula  for  the  radius  of  curvature,         .  154 

Variation  of  the  curvature  at  different  points,       .  .       155 

Radius  of  curvature  for  lines  of  the  second  order,    .  156 

Involute  and  evolute  curves  defined,    .         .         .  .158 

Normal  to  the  involute  is  tangent  to  the  evolute,        .  160 
Difference  between  two  radii  of  curvature  equal  to  the 

intercepted  arc  of  the  evolute,           .         .         .  162 

Equation  of  the  evolute,      .         .         .         .         .  .163 

Evolute  of  the  common  parabola,     .         .         .         .  164 

CHAPTER  VIII. 

Transcendental  curves  defined — Logarithmic  curve,     .       166 

The  cycloid, 169 

Expressions  for  the  tangent,  normal,  &c.,  to  the  cycloid,       171 
Evolute- of  the  cycloid,   .         .         .         .         .         .  173 

Spirals  defined,  .         .         .         .         .         .         .175 


CONTENTS. 


INTEGRAL  CALCULUS. 

Png* 

Integral  calculus  defined,    .         .         .                  .  189 

Integration  of  monomials,         .....  190 

Integral  of  the  product  of  a  differential  by  a  constant,  192 

Arbitrary  constant,           .         .         .         .         .         .  194 

Integration  —  when  a  logarithm,   ...         .  •      '.         .  1  94 

Integration  of  particular  binomials,           .         .         .  195 

Integration  —  when  a  logarithm,   .         .         .    t  .  ..         .  195 

Integral  of  the  differential  of  an  arc  in  terms  of  its 

sine,  .......  "...'.'  196 

Integral  of  the  differential  of  an  arc  in  terms  of  its 

cosine,        .         .         .                  .         .         .         .  198 

Integral  of  the  differential  of  an  arc  in  terms  of  its 

tangent,       .....         .         ...  199 

Integral  of  the  differential  of  an  arc  in  terms  of  its 

versed-sine,                   .         .         .         .         .         .  200 

Integration  by  series,  .  .  .  ...  201 

"  of  differential  binomials,  .  .  .  .  207 

Formula  for  diminishing  the  exponent  of  the  parenthesis,  212 

Formulas  for  diminishing  exponents  when  negative,  .  213 
Particular  formula  for  integrating  the  expression 

214 

—-—-'+                  •                 .                 •                 •                 ,                 •                 •  /OJ.H 


Integration  of  rational  fractions  when  the  roots  of  the 

denominator  are  real  and  equal,  .  .  .  216 
Integration  of  rational  fractions  when  the  roots  are 

equal,     .  ......  221 

Integration  of  rational  fractions  when  the  denominator 

contains,  imaginary  factors,  ....  226 

Integration  of  irrational  fractions,  ....  233 

Rectification  of  plane  curves,  ....  243 

Quadrature  of  curves,  ......  250 

"          of  curved  surfaces.         .         .         .         .  261 

Cubature  of  solids,     .......  269 

Double  integrals,  .  .  .  ,  ;r  274 


DIFFERENTIAL  CALCULUS. 


CHAPTER   I. 
Definitions  and  Introductory  Remarks. 

1.  All  the  quantities  which  are  considered  in  the  Dif- 
ferential Calculus  may    be   divided  into    two  principal 
classes  :  constants  and  variables.     Each  constant  retains 
the  same  value  throughout  the  same  investigation ;  but 
the  variable  quantities  are  subjected  to  certain  laws  of 
change,  in  consequence  of  which  they  may  assume  in  suc- 
cession, an  infinite  number  of  different  values,  without 
changing  the  form  of  the  expression  into  which  they  enter. 

The  constant  quantities  are  generally  designated  by  the 
first  letters  of  the  alphabet,  a,  6,  c,  &c. ;  and  the  variable 
quantities  by  the  final  letters,  x,  y,  z,  &c. 

2.  If  two  variable  quantities  are  so  connected  together 
that  any  .change  in  the  value  of  the  one  necessarily  pro- 
duces a  change  in  the  value  of  the  other,  they  are  said  to 
be  functions  of  each  other. 

Thus,  in  the  expression 

y  =  <wr, 


10  ELEMENTS    OF    THE 

y  and  x  are  functions  of  each  other ;  for,  if  any  change 
be  made  in  the  value  of  x,  a  corresponding  change  will 
take  place  in  that  of  y ;  and  reciprocally. 

3.  When  the  value  of  one  variable  depends  on  that  of 
another,  as  in  the  expression 

y  =  ax,        or        y  =  co?2, 

if  we  attribute  at  pleasure  any  increment  to  one  of  the 
variables,  a  corresponding  change  will  take  place  in  the 
other ;  and  hence,  if  one  of  them  be  supposed  to  increase 
or  decrease  according  to  any  independent  or  arbitrary 
law,  a  corresponding  change  of  the  other  will  take  place 
according  to  the  law  of  relation  which  exists  between 
them.  The  one  to  which  the  arbitrary  increment  is 
given,  is  called  the  independent  variable,  or  simply  the 
variable,  and  the  other  is  called  the  function. 

4.  The  relation  between  a  function  and  its  variable  is 
generally  expressed  thus : 

y=/0), 

in  which  /  is  a  mere  symbol,  denoting  that  y  and  x  are 
functions  of  each  other.  The  expression  is  read,  y  a 
function  of  x,  or  y  equal  to  a  function  of  x. 

The   mutual  dependence  of    one  variable  on  another 
may  also  be  expressed  under  the  form 

f(x,y)  =  0 
in  which  y  is  a  function  of  x,  and  x  a  function  of  y. 

5.  Functions  are  either  increasing  or  decreasing.     An 
increasing  function  is  that  which  increases  when  its  vari- 
able increases,  and  decreases  when  its  variable  decreases 


DIFFERENTIAL    CALCULUS.  11 

Thus,  in  the  expressions 

y  =  ax2,         u  =  (a  +  x)2, 

y  andw  are  increasing  functions  of  x  ;  since,  if  x  be  in- 
creased,?/ and  u  will  both  increase  ;  and  if  a?  be  diminish- 
ed y  and  u  will  both  decrease. 

A  decreasing  function  is  that  which  increases  when  its 
variable  decreases,  and  decreases  when  its  variable  in- 
creases. Thus,  in  the  expression 


y  is  a  decreasing  function   of  x  ;  for,  if  x  is  decreased 
y  will  increase  ;  and  reciprocally. 
In  the  expression 

y  =  (a-^, 

y  will  decrease  while  x  increases  between  the  limits  of 
zero  and  a  ;  but  will  increase  with  x  for  all  values  of  x 
greater  than  a.  Hence,  y  is  a  decreasing  function  of  x 
for  all  values  of  x  less  than  a,  and  an  increasing  func- 
tion of  x  for  all  values  of  x  greater  than  a. 

6.  Functions  are  either  explicit  or  implicit.  An  ex- 
plicit  function  is  when  the  value  of  the  function  is  di- 
rectly expressed  in  terms  of  the  variable  on  which  it  de- 
pends. Thus,  in  the  expressions 

u  =  bxz,        y  =  Va2  —  x2, 
u  and  y  are  explicit  functions  of  x. 

An  implicit  function  is  one  where  the  value  is  not 
directly  expressed  in  terms  of  the  variable.  Thus,  in 
the  expressions 


12  ELEMENTS    OF    THE 

au2  +  cx2  =  bo?,         y2  +  x  =  a2  —  x2, 
u  and  y  are  implicit  functions  of  x.     These  expressions 
may  be  written  under  the  form 

J[u&)  =  0,         and        /(*/,*)  =  0. 
The  relation  between  an  implicit  function  and  its  variable 
may  also  be  expressed  by  means  of  two  or  more  equa- 

tions.    Thus,  if  we  have 

• 

z  =  ay2,         and        y  =  ax, 

z  is  an  implicit  function  of  x. 

These  expressions  may  be  written 

and  =x' 


or  we  may  write 

/(*#)  =  0,         and        /(*/,*)  =0. 

7.  Functions  are  either  algebraic  or  transcendental. 

8.  An  algebraic  function  is  one  in  which  the  relation 
between  it  and  its  variable  can  be  expressed  algebraically 
—  that  is,  by  addition,  subtraction,  multiplication,  division, 
or  the  extraction  of  roots  indicated  by  constant  indices. 
Thus,  in  the  expressions 


u  =  ax2  +  ex,        y  =  -/a2  —  x2, 
u  and  y  are  algebraic  functions  of  x. 

9.  Transcendental  functions  are  those  in  which  the 
relation  between  the  function  and  its  variable  cannot  be 
determined  by  methods  purely  algebraic.  For  example  : 

u  =  ax,         u  —  log.  x,        u  =  sin  x, 
are  transcendental  functions. 


DIFFERENTIAL    CALCULUS.  13 

When  the  variable  enters  as  an  exponent,  it  is  called 
an  exponential  function ;  when  it  enters  as  a  logarithm, 
it  is  called  a  logarithmic  function;  and  when  it  enters  as 
a  sin.,  tang.,  cos.,  &c.,  it  is  called  a  circular  function. 
Thus,  in 

u  —  ax,  u  is  an  exponential  function  of  x ; 

u  =  logo?,  u  is  a  logarithmic  function  of  x ; 

u  =  sin  x,  u  is  a  circular  function  of  x. 

10.  Although  the  values  of  the  function  and  variable 
may  be  changed  at  pleasure  without  affecting  the  values 
of  the  constants  with  which  they  are  connected,  there  is, 
nevertheless,  a  relation  between  them  and  the  constants 
which  it  is  important  to  consider. 

If,  in  the  equation 

y=fa\ 

a  particular  value  be  attributed  either  to  x  or  y,  the  otner 
will  be  expressed  in  terms  of  this  value  and  the  constant 
quantities  which  enter  into  the  primitive  equation.  Thus, 
in  the  equation 

y  —  ax  +  b, 

if  a  particular  value  be  attributed  to  x,  the  corresponding 
value  of  y  will  depend  on  the  value  assigned  to  x,  and  on 
a  and  6;  or  if  a  particular  value  be  attributed  to  y,  the 
corresponding  value  of  x  will  depend  on  that  value,  and 
on  a  and  b.  The  same  will  evidently  be  the  case  in  the 
equation 


14  ELEMENTS    OF    THE 

or  in  any  equation  of  the  form 

y  =--/!». 

Hence,  we  see  that,  although  the  changes  which  take 
place  in  the  values  of  the  function  and  variable  are  entire- 
ly independent  of  the  constants  with  which  they  are  con- 
nected, yet  their  absolute  values  are  dependent  on  those 
constants. 

11.  Since  the  relations  between  the  variables  and  con- 
stants are  not  affected  by  the  changes  of  value  which  the 
variables  may  experience,  it  follows  that,  if  the  constants 
be  determined  for  particular  values  of  the  variables,  they 
will  be  known  for  all  others. 

Thus,  in  the  equation 

a?2  +  y1  =  Rz, 
if  we  make  o?=0,  we  have 

y=±R; 
or,  if  we  make  y=0,  we  have 

x=±R. 

12.  The  function  y,  and  the  variable  oc,  may  be  so  re- 
lated  to  each  other  as  to  reduce  to  0  at  the  same  time 
Thus,  in  the  equation 


which  may  be  placed  under  the  general  form 
/(#,*/)  =  0,         or         y=/0*0, 

if  we  make  x  =  0,  we  have  y  =  0,  or  if  we  make  y  =  0, 
we  shall  have  x  =  0. 


DIFFERENTIAL    CALCULUS.  15 

13.  We  have  thus  far  supposed  the  function  to  depend 
on  a  single  variable  ;  it  may  however  depend  on  several. 
Let  us  suppose,  for  example,  that  u  depends  for  its  value 
on  x,  y,  and  z  ;  we  express  this  dependence  by 

«  =/(*>  y>  *)• 

If  we  make  #—  0,  we  have 


if  we  also  make  y==0,  we  have 

u  =  f(z  ; 

and  if,  in  addition,  we  make  z  =  0,  we  have 
u  —  a  constant, 

which  constant  may  itself  be  equal  to  0. 
If  the  function  be  of  the  form 

u  =  b  +  ax  +  yx2  +  zx2, 

in  which  one  of  the  variables  is  a  factor  of  several  of  the 
terms,  then,  if  x=  0,  we  shall  have 

u=l; 

or,  if  x  were  a  factor  of  all  the  terms,  we  should  have, 
for  x  =  0,         u  =  0. 

14  Let  us  now  examine  the  change  which  takes  place 
in  the  function,  in  consequence  of  any  change  that  may 
be  made  in  the  value  of  the  variable  on  which  it  depends. 

Let  us  take,  as  a  first  example, 

u=ax2,     (1) 
and  then  suppose  x  to  be  increased  by  any  quantity  h. 


16  ELEMENTS    OP    THE 

Designate  by  -uf  the  new  value  which  u  assumes,  under 
this  supposition,  and  we  shall  have 

u'=a(x+h)2, 
or,  by  developing, 

u'  =  ace2-  +  2axh  +  ah2. 

If  we  subtract  the  first  equation  from  the  last,  we  shall 
have 

u'  —  u  —  2axh  +  ah2  ; 

hence,  if  the  variable  oc  be  increased  by  h,  the  function 
will  be  increased  by  2axh  +  ah2. 

If  both  members  of  the  last  equation  be  divided  by  h, 
we  shall  have 

u/  — u      o  r 
=  2ax  +  ah, 

h 

which  expresses  the  ratio  of  the  increment  of  the  variable 
to  that  of  the  function. 

Let  us  take,  as  a  second  example, 
u  =  x*,     (2) 

and  suppose  x  to  be  increased  by  a  quantity  h;  desig- 
nating by  u'  the  new  value  which  u  assumes  under  this 
supposition,  and  we  shall  have 

u'  =  (x  +  h)*, 
and,  by  developing, 

uf  =  x3  +  3x2h  +  3z7i2  +  h\ 

By  transposing  x3,  substituting  for  it  its  value  u,  and 
then  "dividing  by  h,  we  have 


DIFFERENTIAL    CALCULUS.  17 


From  equation  ( 1 )  we  have 
u'  —  u 


=  2ax  +  ah  ; 
h 


and  from  equation  (2), 


h 

15.  Let  us  now  observe  that  the  numerator  in  the  first 
member  of  each  of  the  above  equations,  is  the  difference 
between  the  primitive  function  u,  and  the  new  value  u'  , 
which  arose  from  giving  an  increment  h  to  the  variable 
x,  of  which  u  is  a  function.  Hence  we  see,  that  the  first 
member  of  each  equation  is  equal  to  the  increment  of 
the  function  divided  by  the  corresponding  increment  of 
the  variable. 

If  we  examine  the  second  members  of  these  equations, 
we  find  a  term  in  each  which  does  not  contain  the  in- 
crement h,  viz.  :  in  the  first,  the  term  2ax,  and  in  the 
second,  3x2.     If  now,  we  suppose  h  to  diminish,  it  is 
evident  that  the  terms  2ax,  and  3x2,  which  do  not  contain 
h,  will  remain    unchanged,  while  all    the  terms    which 
contain  h  will  diminish.     Hence,  the  ratio 
u'  —  u 
~hT 

in  either  equation,  will  change  with  h,  so  long  as  h  re- 
mains in  the  second  member  of  the  equation  ;  but  of  all 
the  ratios  which  can  subsist  between 
u'  -u 
~T~' 

is  there  one  which  does  not  depend  on  the  value  of  h? 
We  have  seen  that  as  h  diminishes,  the  ratio  in  the  first 

2 


18  ELEMENTS    OP    THE 

equation  approaches  to  2ax,  and  in  the  second  to  3z2 ; 
hence,  2ax  and  3x2,  are  the  limits  towards  which  the 
ratios  approach  in  proportion  a  h  is  diminished ;  and 
hence,  each  expresses  that  particular  ratio  which  is  in- 
dependent of  the  value  of  h.  This  ratio  is  called  the 
limiting  ratio  of  the  increment  of  the  variable  to  the 
corresponding  increment  of  the  function. 

16.  We  are  now  to  explain  the  notation  by  means  of 
which  this  limiting  ratio  is  to  be  expressed.  For  this 
purpose  let  us  resume  the  equation 


h 
and  represent  by  dx  the  last  value  of  7z,  that  is,  the  vi 

of  h,  which  cannot  be  diminished,  according  to  the  law  of 
change  to  which  h  or  x  is  subjected,  without  becoming  0 
and   let  us  also  represent  by  du  the  corresponding  dif- 
ference between  w'and  u;  we  then  have 


dx. 
The  letter  d  is  used  merely  as  a  characteristic,  and  the 

expressions  du,  dx,  are  read,  differential  of  u,  differential 

of  a:. 

It  may  be  difficult  to  understand  why  the  value  which 

h  assumes  in  passing  to  the  limiting  ratio,  is  represented 
by  dx  in  the  first  member,  and  made  equal  to  0  in  the 
second.  We  have  represented  by  dx  the  last  value  of  h, 
and  this  value  forms  no  appreciable  part  of  h  or  x.  For, 
if  it  did,  it  might  be  diminished  without  becoming  0,  and 
therefore  would  not  be  the  last  value  of  h.  By  designa- 
ting this  last  value  by  dx,  we  preserve  a  trace  of  the 


DIFFERENTIAL    CALCULUS.  19 

letter  #,  and  express  at  the  same  time  the  last  change 
which  takes  place  in  h,  as  it  becomes  equal  to  0.  For  a 
like  reason  the  last  difference  between  u!  and  u  is  desig- 
nated by  du. 

The  limiting  ratio  in  equation  (2)  is 

^  =  3**. 

dx 

The  limiting  ratio  of  the  increment  of  the  variable  to 
that  of  the  function,  which  has  been  found  in  the  pre- 
ceding equations,  is  called  the  differential  coefficient  of  M 
regarded  as  a  function  of  x. 

17.  Let  us  take,  as  another  example,  the  function 

u  =  ax*  :     (3) 
if  we  give  to  x  an  increment  h,  we  shall  have 

u'  =  ax*  +  4ax^h  +  6ao:2/i2  -f-  4axh3  +  ah*,     and 


—  r^  =  4a*3  +  Gax*h  +  *ah*  +  ah3, 
h 

and,  by  taking  the  limiting  ratio,  we  have  for  the  differ- 

ential coefficient, 

du  .» 

—  =  4aar*. 

dx 

18.  If  it  were  required  to  find  the  differential  of  the 
function  u,  after  we  had  formed  its  differential  coefficient, 
it  could  be  done  by  simply  muliplying  the  differential 
coefficient  by  the  differential  of  the  variable  ;  thus,  from 
equation  (l)  we  should  have 

du—  Zaxdx  ; 
from  the  second, 


20  ELEMENTS    OF    THE 

and  from  the  third, 


du  = 

The  differential  of  each  function   may  also  be  written 
under  the  following  form  : 

Eq.  1,  --dx  =  2axdx  ; 


Eq.  2, 


du 

Eq.  3,  -r-dx  =  4ax3dx; 

which,  indeed,  is  nothing  more  than  finding  the  differen- 
tial of  the  function  by  multiplying  the  differential  coefficient 
expressed  in  the  first  member  of  the  equation,  by  the  dif- 
ferential of  the  variable. 

19.  Let  us  now  examine  each  of  the  three  equations 
which  we  have  considered,  and  observe  the/orwi  of  the 
expression  for  the  difference  between  the  two  states  of 
the  function  ti. 

From  the  first  equation  we  have 

u'  —  u  =  2axh  +  ah2; 
from  the  second, 


and  from  the  third, 

u'  -  u  =  4.ax3h  +  6ax*hz  +  4a*7i3  +  a/*4. 
We  see  in  each  of  the  expressions  for  the  difference 
between  the  two  states  of  the  function  u,  that  the  first 
term  of  the  difference  contains  the  first  power  of  the  in- 
crement /*,  and  that  the  coefficient  of  this  term  is  the 
differential  coefficient  of  the  function  u,  or  the  limiting 


DIFFERENTIAL    CALCULUS.  21 

ratio  of  the  increment  of  the  function  to  that  of  the  vari- 
able;   This  differential  coefficient  is,  in  general,  a  func 
tion  of  x. 

If,  now,  in  either  of  the  expressions,  we  represent  the 
differential  coefficient,  or  limiting  ratio,  by  P,  and  all  the 
following  terms  of  the  difference  by  P'h2  (in  which  P 
will  in  general  be  a  function  of  h),  the  difference  may  be 
written  under  the  form 

u'  -u  =  Ph  +  P'ha; 

and  we  shall  assume  that  what  has  been  proved  in  regard 
to  the  three  forms  of  the  function  u  which  we  have  con- 
sidered, is  equally  true  for  all  other  forms.  This  form, 
for  the  difference  between  the  two  states  of  the  function, 
is  important,  and  should  be  carefully  remembered.  If, 
then,  we  have  a  function  of  the  form 

«  =/(*), 

and  give  to  x  an  increment  hf  we  shall  have 

iS-u=  Ph  +  P'h\ 
If,  now,  we  wish  the  ratio  of  the  increments,  we  have 


and,  passing  to  the  limiting  ratio, 


and,  if  we  wish  the  differential  of  the  function  w,  we  have 
du  =  Pdx, 

du, 

or  -rdx  =  Pdx. 

dx 

If  we  represent  the  increment  of  the  variable  by  k,  and 


22  ELEMENTS    OF    THE 

the  differential  coefficient  by  Q,  the  difference  would  be 

represented  by 

u'  -  u  =  Qk  +  Q'ka 

du 
and  fadx  =  Qdx. 

We  may  conclude  from  the  above,  that  if  we  have  the 
difference  between  two  states  of  a  function,  as 

vf  -  u  =  Ph  +  P'h\ 

that  we  can  immediately  pass  to  the  differential  ofu,by 
writing  du  for  u'—  u,  substituting  dxfor  h  in  the  second 
member,  and  omitting  the  terms  which  contain  ha. 

20.  If  two  functions,  u  and  v,  dependent  on  the  same 
variable,  are  equal  to  each  other,  for  all  possible  values 
which  may  be  attributed  to  that  variable,  the  differentials 
of  those  functions  will  also  be  equal. 

For,  suppose  x  to  be  the  independent  variable.  We 
shall  then  have  (Art.  15), 


v'-v  =  Qh  +  Q'h2, 

in  which  Q  is  the  differential  coefficient  of  v,  regarded  as 
a  function  of  #. 

But,  since  w'and  v'  are,  by  hypothesis,  equal  to   each 
other,  as  well  as  u  and  v,  we  have 

Ph  +  P'h2  =Qh+  Q?V, 
or,  by  dividing  by  h  and  passing  to  the  limiting  ratio, 

P  =  Q, 

du      dv 

hence'  55  =  5? 


DIFFERENTIAL    CALCULUS.  23 

du  dv 

and  -j-  dx  —  -j-dx, 

dx  dx 

that  is,  the  differential  of  u  is  equal  to  the  differential  of  v. 

21.  The  reverse  of  the  above  proposition  is  not  gene- 
rally true ;  that  is,  if  two  differentials  are  equal  to  each 
other,  we  are  not  at  liberty  to  conclude  that  the  functions 
from  which  they  were  derived  are  also  equal. 

For,  let 

in  which  A  is  a  constant,  and  u  and  v  both  functions  of 
x.  Giving  to  x  an  increment  A,  we  shall  have 

u'  =  v'±A, 

from  which  subtract  the  primitive  equation,  and  we  ob- 
tain 

u'  —  u  =v/  —  v, 

and,  by  substituting  for  the  difference  between  the  two 
states  of  the  function,  we  have 

Ph  +  P'h2=Qh+Qfh2 

Dividing  by  A,  and  passing  to  the  limiting  ratio,  we 
obtain 

du      dv 

P  =  Q;     that  is      -y-  =  -y-  ; 
dx      dx' 

du  .       dv _ 

hence,  -y-  ax=  -r-aff  ; 

dx          dx 

or,  what  is  the  same  thing,  by  merely  changing  the  form, 

du  =  dv. 

Here  we  see  that  although  v  may  be  greater  or  less  than 
u  by  the  constant  quantity  A,  still  its  differential  will 
always  be  equal  to  that  of  u. 


24  ELEMENTS    OF    THE 

Hence,  also,  we  conclude  that  every  constant  quantity, 
connected  with  the  variable  by  the  sign  plus  or  minus, 
will  disappear  in  the  differentiation. 

The  reason  of  this  is  apparent;  for,  as  a  constant  ad- 
mits of  no  increase  or  decrease,  there  is  no  ultimate  or 
last  difference  between  two  of  its  values ;  and  this  ulti- 
mate or  last  difference  is  the  differential  of  a  variable 
function.  Hence  the  differential  of  a  constant  quantity 
is  equal  to  0. 

22.  If  we  have  a  function  of  the  form 

u  —  Av, 

in  which  u  and  v  are  both  functions   of  x,  and  give  to  x 
an  increment  h,  we  shall  have 

u'  —  u  —  A(v'  —  v\ 

or  Ph  +  P'tf  =  A(Qh+  Q'hz) ; 

and,  by  dividing  by  h,  and  passing  to  the  limiting  ratio, 

we  have 

P  =  AQ9 

or  Pdx  =  AQdx. 

But  Pdx  =  du,     and     Qdx  =  dv9 

hence,  du  —  Adv  ; 

that   is,  the  differential  of  the  product  of  a   variable 

quantity  by  a  constant,  is  equal  to  the  differential  of  the 

variable  Multiplied  by  the  constant. 


DIFFERENTIAL    CALCULUS.  25 


CHAPTER  II. 

Differentiation    of  Algebraic   Functions — Succes- 
sive   Differentials — Taylor's  and   Madauriris 

Theorems. 

+ 

23.  Algebraic  functions  are  those  which  involve  the  sum 
or  difference,  the  product  or  quotient,  the  roots  or  powers, 
of  the  variables.     They  may  be  divided  into  two  classes, 
real  and  imaginary. 

24.  Let  it  be   required  to  find  the  differential  of  the 
function. 

u  =  ax. 

If  we  give  to  x  an  increment  A,  and  designate   the 
second  state  of  the  function  by  uf,  we  shall  have 

uf  =  ax  +  ah  =  u  +  ah, 
u'-u 

-JT  =  a: 

hence,          du  —  adx,      or      —j—dx  =  adx. 

dx 

25.  As  a  second  example,  let  us  take  the  function 

u  =  ax*. 


26  ELEMENTS    OF    THE 

If  we  give  to  x  an  increment  h,  we  have 
u'  =  ax*  +  2  ahx  -f  ah9, 


hence,  du  =  2  ax  dx. 

26.  For  a  third  example,  take  the  function 

u  =  ax3  : 
giving  to   x   an  increment  h,  we  have 


h 
or  passing  to  the  limit 

-1—  =  3  ao?2  ;     hence,     du  =  3  ax2  dx. 
dx 

27.  Let  us  now  suppose  the  function  u  to  be  composed 
of  several  variable  terms  :  that  is,  of  the  form 

u  —  y-\-z  —  w  —  f(x\ 

in  which   y,  z,  and  w,  are  functions  of  x. 

If  we  give  to  x  an  increment  h,   we  shall  have 

</_  U  =  (y>  _  y)  +  (/_  ^  )  -  („,'_  W)  : 

hence,  (Art.  19), 


or, 

/i 

or  by  passing  to  the  limit 

-^ 
dx 


DIFFERENTIAL    CALCULUS.  27 

and  multiplying  both  members  by  dx,  we  have 


ax 

But  since  P,  Q,  and  L,  are  the  differential  coefficients 
of  y,  z,  and  to,  regarded  as  functions  of  x,  it  follows  (Art. 
18)  that,  the  differential  of  the  sum,  or  difference  of  any 
number  of  functions,  dependent  on  the  same  variable,  is 
equal  to  the  sum  or  difference  of  their  differentials  taken 
separately. 

28.  Let  us  now  determine  the  differential  of  the  product 
of  two  variable  functions. 

If  we  designate  the  functions  by  u  and  v,  and  suppose 
them  to  depend  on  a  variable  x,  we  shall  have 


and  by  multiplying 

wV  =  (u  +  Ph  +  P'h2)  (v  +  Qh  + 

=  uv  +  vPh  +  uQh  +  PQh2  +  &c  ; 
hence 

u'v'  —  uv 

-  -  -  =  vP  +  wQ-f  terms  containing  h,  hz,  &  h\ 

If  now,  we  pass  to  the  limiting  ratio,  we  have 

d(uv)        n  ,  > 

-±-J.  =  vP  +  uQ; 
dx 

therefore,  d(uv)  =  vPdx  +  uQdx  =  vdu  +  udv. 

Hence,  the  differential  of  the  product  of  two  functions 
dependent  on  the  same  variable,  is  equal  to  the  sum  of  the 


28  ELEMENTS    OF  THE 

products  which  arise  by  multiplying  each  by  the  differ 
ential  of  the  other. 

If  we  divide  by  uv,  we  have 

d(uv)      du      dv 
uv         u        v  ' 

that  is,  the  differential  of  the  product  of  two  functions,  di- 
vided by  the  product,  is  equal  to  the  sum  of  the  quotients 
which  arise ,  by  dividing  each  differential  by  its  function. 
29.  We  can  easily  determine    from   the   last  formula, 
the  differential  of  the  product  of  any  number  of  functions. 
For  this  purpose,  put  v  =  ts,   then 

dv  _  d(ts)  _ _dt_       ds 
~v~ '        ts         ~T  '  '  ~s~' 

and  by  substituting  for  v   in  the  last  equation,  we  have 

d(uts)        du    .    dt        ds 
uts  u          t          s 

and  in  a  similar  manner,  we  should  find 

d(utsr )=  du    j    dt    }    ds  ^  dr  ^         &^ 

utsr ....         u          t          s         r 

If  in  the  equation 

d(uts)  _  du        dt        ds 
uts  ~     u          t          s  ' 

we  multiply  by  the  denominator  of  the  first  member,  we 
shall  have 

d(uts)  =  tsdu  +  usdt  +  utds  ; 

and  hence,  the  differential  of  the  product  of  any  number 
of  functions,  is  equal  to  the  sum  of  the  products  which 


DIFFERENTIAL    CALCULUS. 


arise  by  multiplying  the  differential  of  each  function  by 
the  product  of  all  the  others. 

30.  To  obtain  the  differential  of  any  fraction,  as 


—     we  make 


—  =  t.     and  hence     u  =  tv. 
v 


Differentiating  both  members,  we  find 

du  —  vdt  +  tdv ; 
finding  the  value  of  dt,  and  substituting  for  t  its  value 


u  .     . 

— ,     we  obtain 
v 


j.      du       udv 

at  = — , 

v          v2 


or  by  reducing  to  a  common  denominator 
,  vdiif  —  udv 

~~s~ 

hence,  the  differential  of  a  fraction  is  equal  to  the  deno- 
minator into  the  differential  'of  the  numerator,  minus  the 
numerator  into  the  differential  of  the  denominator,  divided 
by  the  square  of  the  denominator. 

31.  If  the  numerator  u  is  constant  in  the  fraction  t  =  — , 

v 

its  differential  will  be   0  (Art.  21),  and  we  shall  have 

,  udv  dt  u 

dt= 2~,         or        -y-= 5. 

v2  dv  v2 

When  u  is  constant,  t  is  a  decreasing  function  of  v  (Art. 
5),  and  the  differential  coefficient  of  t  is  negative. 

This  is  only  a  particular  case  of  a  general  proposition 


30  ELEMENTS    OP    THE 

For,  let  u  be  a  decreasing  function  of  x.     Then,  if  we 
give  to   x  any  increment,  as        h,   we  have 


or  vf-u  =  Ph-t  Pfh\ 


But  by  hypothesis  u>uf  \  hence,  the  second  member 
is  essentially  negative  for  all  values  of  h  ;  and,  passing 
to  the  limiting  ratio, 

^-    _P 

dx~ 

hence,  the  differential  coefficient  of  a  decreasing  function 
is  negative. 

32.  To  find  the  differential  of  any  power  of  a  function, 
let  us  first  take  the  function  wn,  in  which  n  is  a  positive 
and  whole  number.  This  function  may  be  considered  as 
composed  of  n  factors  each  equal  to  u.  Hence,  (Art.  29), 

d(un)  _  d(uuuu  .  .  .  .  )  _  du       du      du      du 
un  ~      (uuuu  .  .  .  .  )  ~  u        u        u        u 


But  since  there  are  n.  equal  factors  in  the  first  member, 
there  will  be  n  equal  terms  in  the  second  J  hence, 

d(un)  _  ndu  m 
un  ^      u    ' 

therefore,  d  (un)  =  nun~l  du. 

If  n  is  fractional,  represent  it  by  — ,  and  make 

o 

r^ 

v  =  u',       whence,        ur  =  v'; 

and  since  r  and  s  are  supposed  to  represent  entire  num- 
bers, we  shall  have 

rur~*  du  =  sv'~ldv  ; 


DIFFERENTIAL    CALCULUS.  31 

from  which  we  find 

dv  -  n/r~1  du  -    rM<"1    du 

su~*(" 
or  by  reducing 

dv  =' — u'      du; 
which  is  of  the  same  form  as  the  function 


V 

by  substituting  the  exponent  —  for  n. 

s 

Finally,  if  n  is  negative,  we  shall  have 

1 


from  which  we  have  (Art.  31), 


u 
hence,  by  reducing 

d(u-n)=-nu-n-ldu. 

Hence,  the  differential  of  any  power  of  a  function,  is 
equal  to  the  exponent  multiplied  by  the  function  with  its 
primitive  exponent  minus  unity,  into  the  differential  of  the 
function. 

33.  Having  frequent  occasion  to  differentiate  radicals  of 
the  second  degree,  we  will  give  a  specific  rule  for  this 
class  of  functions. 

Let  v  =  ^fu^       or        v  =  wr; 

1    4~»  ,        1    -4  ,  du 

then,          dv  —  ~ u*     du  —  -^u   *du  = 


32  ELEMENTS    OF   THE 

that  is,  the  differential  of  a  radical  of  the  second  degree, 
is  equal  to  the  differential  of  the  quantity  under  the  sign, 
divided  by  twice  the  radical. 

34.  It  has  been  remarked  (Art.  3),  that  in  an  equation 
of  the  form 

u  =  f(x), 

we  may  regard  u  as  the  function,  and  x  as  the  variable, 
or  x  as  the  function,  and  u  as  the  variable.  We  will 
now  show  that,  the  differential  coefficient  which  is  obtained 
by  regarding  u  as  a  function  of  x,  is  equal  to  the  recip- 
rocal of  that  which  is  obtained  by  regarding  x  as  a  func- 
tion of  u. 
If  we  have 


and  give  to  x  an  increment  h,  we  have  (Art.  19), 

u'-u^Ph  +  P'h2.     (1) 
But,  if  x  be  expressed  in  u,  and  we  have 

*  =/(«)> 

and  then  give  to  u  an  increment  k,  we  shall  nave 
aS-x  =  h=:Qk  +  Q'k*.     (2) 

But  k  =  u'  —  u.     Substituting  these  values  for  u'  —  u, 
and  h,  in  equation  (l),  and  we  have 

k  —  PQk  +  terms  containing  the  higher  powers  of  k. 
Dividing  by  k,  and  passing  to  the  limiting  ratio,  we  have 

1=PQ,         or         P  =      . 


DIFFERENTIAL    CALCULUS.  33 

To  illustrate  this  by  an  example,  let 

j 
u  =  a?,     whence     x  =  tyu^=  u  3  • 


Now,  -- 

dx 


but  regarding  x  as  the  function 
dx 


35.  If  we  have  three  variables  u,  y,  and  #,  which  are 
mutually  dependant  on  each  other,  the  relations  between 
them  may  be  expressed  by  the  equations 

u=f(y\         and         y=f'(x). 

If  now  we  attribute  to  x  an  increment  h,  and  designate 
by  k,  the  corresponding  increment  of  y,  we  shall  have 


(Art.  19), 


and 


If  we*  multiply  these  equations  together,  member  by 
member,  we  shall  have 


but  li  —  y'  —  y;  hence,  by  dividing  and  passing  to  the 
limiting  ratio,  we  have 

du       du        dv 

-7-  =  T~  x     7     > 
dx       ay        dx 

and  hence,  if  three  quantities  are  mutually  dependant  on 

3 


34  ELEMENTS    OF    THE 

each  other,  the  differential  coefficient  of  the  first  regarded 
as  a  function  of  the  third,  will  be  equal  to  the  differential 
coefficient  of  the  first  regarded  as  a  function  of  the  second, 
multiplied  by  the  differential  coefficient  of  the  second  re- 
garded as  a  function  of  the  third. 

36.  Let  us  take  as  an  example 


v  =  bur,        u  =  oar. 

we  find 

dv  _    ,  2  du 

du  dx 

dv       dv       du 

But,          — -=— -X-T-  — 36w2x  2ax  = 
ax       du       dx 

and  by  substituting  for  w2,  its  value  a2x*t 
dx  ~ 

EXAMPLES. 


1.  Find  the  differential  of  u  in  the  expression 

i^ 

Put  a2  —  a?  =  y,  then  w  =  y2,  and  the  dependence  be- 
tween u  and  x,  is  expressed  by  means  of  y,  and  u  is 
an  implicit  function  of  x.  Differentiating,  we  find 

du  1     -         1/  ~  d 


DIFFERENTIAL    CALCULUS.  35 

by  multiplying  the  coefficients  together  we  obtain 
du  Ixo        o.-i  _  —  x 


hence, 


f?    _ 

r 


2.  Find  the  differential  of  the  function 

u  =  (a  +  bxn)m. 
Place       a  +  bxn  =  y:       then       u  =  ym;       and 

=  m""1  =  w  a  +  for"" 


-^  =  my""1  =  w  (a 


nence, 


du 


~ 

•j-  =  mnb  (a  +  bx")     a?""1  . 

du  =  mnb  (a  -f  bxn)      x*~  l  dx. 
3.  Find  the  differential  of  the  function 


in  which  the  operations  in  the  last  two  terms  are  only 
indicated.     If  we  perform  them,  we  find 


d(  —  x2)      —  xdx 

^V 


36  ELEMENTS    OF    THE 

Substituting  these  values,  we  find 


or,  reducing  to  a  common  denominator  and  cancelling  die 
like  terms, 


_ 


4.  Find  the  differential  of  the  function 


~ 


_ 


-  a?)  -  (a2  - 


from  which  we  find 


__ 


5.  Find  the  differential  of  the  function 


Make          y  =  ~= 
then  we  shall  have 


DIFFERENTIAL    CALCULUS.  37 

we  therefore  have  (Art.  32), 

3  --i 

du  =  —(a 


_  —  3dy 


But  from  the  equations  above,  we  find 

,         ,/  b  \  d,Tj~x~    —bdx 

d   =  d—  —*—       - 


X  - 


Substituting  these   values   of  </y  and  c?z,   in  the   ex- 
pression for  du,  we  find 


du  = 


38 


ELEMENTS    OF    THE 


8.  u=  vZax  +  x2, 

9  u  = 

10.  u  = 

11.  u  = 


du  = 


(a  +  x)  dx 


,      du  =  6(a2-f        xdx 


18.     tt= 


13.     u  = 


14.     M=L_i 


15.     M  = 


du 


17.     w  = 


4cf  /T      c~H 

^La+v6-dfl 


dx 


\fe-l 


nxn~ldx 


ig   u_ 


. 


DIFFERENTIAL    CALCULUS.  39 

<**(!+  -ST^X*) 


Vl-f^+Vl X  7 

20.  u  =   v,     <  ,  du  =  —  . 

Vl  +  x—Vl  —  x  rfVl—x2 

21.  Find  the  differential  coefficient  of 

Ans.     32^-9^-5. 

22.  Find  the  differential  coefficient  of 


.  Ans. 
23.  Find  the  differential  coefficient  of 


Ans.     2(a 
24.  Find  the  differential  coefficient  of 


F(x)  = 


Of  Successive  Differentials. 

37.  It  has  been  remarked  (Art.  19),  that  the  differ- 
ential coefficient  is  generally  a  function  of  x.  It  may 
therefore  be  differentiated,  and  x  may  be  regarded  as  the 
independent  variable.  A  new  differential  coefficient  may 
thus  be  obtained,  which  is  called  the  second  differential 
coefficient. 


40  ELEMENTS    OF    THE 

38.  In  passing  from  the  function  u  to  the  first  differ- 
ential coefficient,  the  exponent  of  x  in  every  term  in 
which  x  enters,  will  be  changed;  and  hence,  the  rela- 
tion which  exists  between  the  primitive  function  u  and 
the  variable  x,  is  different  from  that  which  will  exist 
between  the  first  differential  coefficient  and  x.  Hence, 
the  same  change  in  x  will  occasion  different  degrees  of 
change  in  the  primitive  function  and  in  the  first  differential 
coefficient. 

The  'second  differential  coefficient  will,  in  general,  be 
a  function  of  x:  hence,  a  new  differential  coefficient 
may  be  formed  from  it,  which  will  also  be  a  function 
of  x ;  and  so  on,  for  succeeding  differential  coefficients. 

If  we  designate  the  successive  differential  coefficients 
by  ' 

p,     q,     r,     s,     &c., 

we  shall  have 

du  dp  da 

-r=p,         J-  =  <1>          i=r>     &c' 
dx  dx  dx 


But  the  differential  of  p   is  obtained  by  differentiating 

its  value     —  ,     regarding  the  denominator    dx    as   con- 
dx 

stant;  we  therefore  have 

du\      j  d2u       , 

d'       or'       •-  =  ** 


and  by  substituting  for  dp  its  value,  we  have 


DIFFERENTIAL    CALCULUS.  41 

The  notation  d2u,  indicates  that  the  function  u  has 
been  differentiated  twice,  and  is  read,  second  differential 
of  u.  The  denominator  da?  expresses  the  square  of  the 
differential  of  x,  and  not  the  differential  of  a?.  It  is 
read,  differential  square  of  x,  or  differential  of  x  squared. 

If  we  differentiate  the  value  of  q,  we  have 

d3u 


hence,  —  j  =  r,     &c., 

and  in  the  same  manner  we  may  find 


The  third  differential  coefficient     -T-^,     is  read,  third 

oar 

differential  of  u  divided  by  dx  cubed;  and  the  differ- 
ential coefficients  which  succeed  it,  are  read  in  a  similar 
manner. 

Hence,  the  successive  differential  coefficients  are 


du  _  d2u  d*u  _ 

~  "***          ~- 


from  which  we  see,  that  each  differential  coefficient  is 
deduced  from  the  one  which  precedes  it,  in  the  same 
way  that  the  first  is  deduced  from  the  primitive  function. 

39.  If  we  take  a  function  of  the  form 
u  =  axn, 


42  ELEMENTS    OP    THE 

we  shall  have  for  the  first  differential  coefficient, 

du 

~  —  nax     . 

dx 

If  we  now   consider    n,    a,    and  dx,  as  constant,  we 
shall  have  for  the  second  differential  coefficient 


2? 

and  for  the  third, 


and  for  the  fourth, 


It  is  plain,  that  when  n  is  a  positive  whole  number,  the 
function 

u  =  axn, 

will  have  n  differential  coefficients.  For,  when  n  dif- 
ferentiations shall  have  been  made,  the  exponent  of  x  in 
the  second  member  will  be  0  ;  hence,  the  nth  differential 
coefficient  will  be  constant,  and  the  succeeding  ones  will 
be  equal  to  0.  Thus, 


n(n-l)(n-2)(n-3)  ......  a.l, 

ax 

n  +  l 
and, 


DIFFERENTIAL    CALCULUS.  43 

Taylor  s  Theorem. 

40.  TAYLOR'S    THEOREM   explains    the   method  of  de- 
veloping into  a  series  any  function  of  the  sum  or  difference 
of  two  variables  that  are   independent   of  each  other,  ac- 
cording to  the  ascending  powers  of  one  of  them. 

41.  Before  giving   the  demonstration  of  this  theorem, 
it  will  be  necessary  to  prove  a  principle  on  which  it  de- 
pends, viz  :  if  we  have  a  function  of  the  sum  or  difference 
of  two  variables 

u  =  f(x±y\ 

the  differential  coefficient  will  be  the  same  if  we  suppose  x 
to  vary  and  y  to  remain  constant,  as  when  we  suppose  y 
to  vary  and  x  to  remain  constant. 

For,  make  x  ±  y  =  z : 

we  shall  then  have 

u=f(z) 

du 

and  -J——P- 

dz 

If  we  suppose  y  to  remain  constant  and  x  to  vary, 
we  have 

dz  =  dx, 

and  if  we  suppose  x  to  remain  constant  and  y  to  vary, 
we  nave 

dz  =  dy. 

But  since  the  differential  coefficient  p  is  independent 
of  dz1  (Art.  15),  it  will  have  the  same  value  whether, 

dz  =  dx,        or,        dz  =  dy. 


44  ELEMENTS    OF    THE 

To  illustrate  this  principle  by  a  particular  example,  let 
us  take 


If  we  suppose   x  to   vary  and  y  to  remain  constant, 
we  find 

du 


and  if  we  suppose   y   to  vary  and   x  to  remain  constant, 
we  find 


the  same  as  under  the  first  supposition. 
42.  It  is  evident  that  the 


must  be  expressed  in  terms  of  the  two  variables  x  and  y, 
and  of  the  constants  which  enter  into  the  function. 
Let  us  then  assume 

f(x  +  y)  =  A  +  By*  +  Cy>  +  .%'  +,  &c, 

in  which  the  terms  are  arranged  according  to-  the  ascend- 
ing powers  of  y,  and  in  which  A,  B,  C,  D,  &c.,  are  inde- 
pendent of  y,  but  functions  of  x,  and  dependant  on  all 
the  constants  which  enter  the  primitive  furiction.  It  is 
now  required  to  find  such  values  for  the  exponents  0,  6,  c, 
&c.,  and  the  coefficients  A,  B,  C,  D,  &c.,  as  shall  ren- 
der the  development  true  for  all  possible  values  which 
may  be  attributed  to  x  and  y. 


DIFFERENTIAL    CALCULUS.  45 

In  the  first  place,  there  can  be  no  negative  exponents 
For,  if  any  term  were  of  the  form 

-    ••     •-;'•;       By-,  '•-.-...••      •: 

it  may  be  written 

.B_ 

y" 

and  making  y  —  0,   this  term  would  become  infinite,  and 
we  should  have 

/(*)=«. 

which  is  absurd,  since  function  of  a?,  which  is  independent 
of  y,  does  not  necessarily  become  infinite  when  y  —  0. 

The  first  term  A,  of  the  development,  is  the  value 
which  the  primitive  function  assumes  when'  we  make 
y  —  0.  If  we  designate  this  value  by  u,  we  shall  have 

/(*)=«. 

If  we  make 


and  differentiate,  under  the  supposition  that  x  varies  and  y 
remains  constant,  we  shall  have 

du'  dA  dB  a  dC  b  ,  dD 
-T-  =  -r--t--7-y+-7-y'4--7- 
dx  dx  djc  dx  dx 

and  if  we  differentiate,  regarding  y  as  a  variable  and  x 
as  constant,  we  shall  find 

-1  +,  &c.  : 

« 

But  these  differential  coefficients  are  equal  to  each  other 
(Art.  41);  hence,  the  second   members  of  the  equations 


46  ELEMENTS    OF    THE 

are  equal,  and  since  the  coefficients  of  the  series  are 
independent  of  y,  and  the  equality  exists  whatever  be  the 
value  of  y,  it  follows  that  the  corresponding  terms  in  each 
series  will  contain  like  powers  of  y,  and  that  the  coef- 
ficients of  y  in  these  terms  will  be  equal  (Alg.  Art.  244). 
Hence, 

a  — 1  =  0,         b  —  l  =  a,         c  —  1=6,     &C., 
and  consequently 

a  =  l,         6  =  2,         c  =  3,     &c.; 
and  comparing  the  coefficients,  we  find 

B  =  —       C=— — —      D=  l   dC 

dx   '  2    dx  '  3    dx 

And  since  we  have  made 
/ 

we  shall  have 

jj      du       n         (Pu          n  d?u 

/7'vi  1     O  /"/'Y^  1      ^      ^  fJ f*^ 

and  consequently, 

.   c?w  d*u      v2     .    d?u     v3 


'  &c- 


43.  This  theorem  gives  the  following  development  for 
the  function 


U  n     l  (U  i  i\      n     I       S 

u  =  a?n,      -3—  =  nxn   ,       -j-f  =n(n  —  I  )xn~lt  &c.  : 
dx  dx* 


•%  DIFFERENTIAL    CALCULUS.  47 

hence, 

*>=(*  +  yY  =  xn  +  nrf-'y  +  fifedfi^i 

n(n-l)(n-2)       y          &c 
•        1.2.3 

44.  The  theorem  of  Taylor  may  also  be  applied  to  the 
development  of  the  second  state  of  any  function  of  the 
form 


when  x  receives  an  arbitrary  increment  h,  and  becomes 
x  +  h.     For,  if  we  substitute  h  for  y,  we  have 

du,      d*u  h2    .    <Pu      A3  . 


v!  —  u      du      d2u    h        d?u      h2 


Now,  it  is   plain  that  h  may  be  made  so  small  that  the 
td2u     1  d3u     h 

K^T^  +  ^Y^+.& 

shall    be  less  than   any   assignable    quantity,  and  conse- 

sequently  less  than  -?f.      Then,    for  any  value  of  h  still 
dx 

smaller,  we  shall  also  have 

du         d2u     1          d3u      h 


or,  if  we  multiply  both  sides  of  the  inequality    by   h,  we 
shall  have 


48  ELEMENTS  OF  THE 

du      <Pu    h2         (Pu     h3 

dx       dx*  1.2   +da?1|1.2.3+' 

that  is,  when  a  series  is  expressed  in  the  ascending 
powers  of  a  variable,  so  small  a  value  may  be  assigned 
to  that  variable  as  shall  render  the  first  term  of  the  series 
greater  than  the  sum  of  all  the  other  terms,  and  this  in- 
equality will  increase  for  all  values  of  the  variable  which 
are  still  less.  Under  such  a  supposition  the  sign  of  the 
series  will  depend  on  that  of  its  first  term.. 

45.  Remark-  The  theorem  of  Taylor  has  been  demor. 
strated  under  the  supposition  that  the  form  of  the  function 


is  independent  of  the  particular  values  which  may  be 
attributed  to  either  of  the  variables  x  or  y.  Hence,  when 
we  make  y  =  0,  and  obtain 

u=f(x), 

this  function  of  x  ought  to  preserve  the  same  form  as 
f(x  -f-  y)  ;  else  there  would  be  values  of  x  in  one  of  the 
functions, 

u'=J[x+y\  «=/(*), 

which  would  not  be  found  in  the  other,  and  consequently 
some   of  the  values  of  x  would  be  made  to  disappear 
when  a  particular  value  is  assigned  to  y,  which  is  entire- 
ly contrary  to  the  supposition. 
If  the  function  be  of  the  form 


u'  =  b  +  Va  —  x  +  y, 
we  shall  have 


DIFFERENTIAL    CALCULUS.  49 

If  we  now  make  x  =  a,  we  shall  have 
u'  =  b  -f- 


in  which  we  see,  that  u'  and  u  are  expressed  under  dif- 
ferent forms  ;  and  hence,  the  particular  value  of  y  —  0 
changes  the  form  of  the  function,  which  is  contrary  to  the 
hypothesis  of  Taylor's  theorem.  When,  therefore,  the 
function 


shall  change  its  form  by  attributing  particular  values  to 
x  or  y,  the  development  cannot  be  made  by  Taylor's 
theorem,  for  such  particular  values. 

46.  The  particular  supposition  which  changes  the  form 
of  the  function  will,  in  general,  render  the  differential 
coefficients  in  the  development  equal  to  infinity. 

If  we  have 


then,  u  =c+ 

du  1 


(Pu 
da? 


rfiii  i      ^ 

Qrlf  1     .    o 

3?^ 


&c.  &c. 

in  which  all  tne  coefficients  will  become  equal  to  infinity 
when  we  make   x  =  — /. 


50  ELEMENTS    OF   THE 

47.  If  we  have  a  function  of  the  form 


u'  =  b  +      a  —  x 

in  which  n   is  a  whole  number,  all  the  differential  coeffi- 
cients of  u,  for  x=a  will  become  infinite.     For,  we  have 


hence, 


u  =  b  4-  tya  —  x  =  b  +  (a  —  a?)" 


&c. 
all  of  which  become  infinite  when  we  make   x  =  a. 

Madauritfs  Theorem. 

48.  MACLAURIN'S   THEOREM   explains   the   method    of 
developing  into  a  series  any  function  of  a  single  variable 
Let  us  suppose  the  function  to  be  of  the  form 


It  is  plain  that  the  value  of  f(x)  must  be  expressed  in 
terms  of  a?,  and  of  the  constants  which  enter  into  f(oc). 
Let  us  therefore  assume 

u  =  A  +  Bxa  +  CV  4-  Dxe  +  ,  &c., 

in  which  the  terms  are  arranged  according  to  the  ascend- 
ing powers  of  x,  and  in  which  A,  B,  C,  D,  &c.,  are 


DIFFERENTIAL    CALCULUS.  51 

independent  of  x,  and  dependent  on  the  constants  which 
enter  into  f(x). 

It  is  now  required  to  find  such  values  for  the  exponents 
G,  b,  c,  &c.,  and  the  coefficients  A,  B,  C,  D,  &c.,  as 
shall  render  the  development  true  for  all  possible  values 
which  may  be  attributed  to  x. 

If  we  make  x  =  0,  u  takes  that  value  which  the  f(x) 
assumes  under  this  supposition,  and  if  we  designate  that 
value  by  U  we  shall  have 

U  =  A. 

The  first  differential  coefficient  is 

• 


+  bCxb~l  +  cDxe~l  +  &c, 
dx 

and  since  this  does  not  necessarily  become  0  when  we 
make  x  =  0,  it  follows  that  there  must  be  one  term  in  the 
second  member  of  the  form  07°  :  hence, 

a  —  1=0,         or         a  =  1  ; 
and  making  x  =  0,   we  have 

^L  =  B=U 

dx 
The  second  differential  coefficient  is 


but  since  the  second  differential  coefficient  does  not  neces- 
sarily become  0,   when  x  =  0,  we  have 

6  —  2  =  0,         or         6  =  2: 


52  ELEMENTS    OP    THE 

hence,  by  making  x  =  0,  we  have 


We  may  prove  in  a  similar  manner  that 

d3u      I  U'' 


c  =  3     and    D  = 


dot?  1.2.3'       1.2.3 


Having  designated  by  U  what  the  function  becomes 
when  we  make  x  =  0,  and  by  W,  U",  Ulff,  &c.,  what 
the  successive  differential  coefficients  become  under  the 
same  supposition,  we  shall  have 

/(«)  =  U  +  Ux  +  U"  ~  +  U"  -£-  +  &c. 

!•«  l«7»*o 

49.  The  theorem  of  Maclaurin  may  be  deduced  imme- 
diately from  that  of  Taylor. 
In  the  development 

du        d2 


_ 

the  coefficients    u,    -^-,        -^,     &c., 
ax  dor 

are  functions  of  x,  and  also  dependent  on  the  constants 
which  enter  into  f(x  +  y). 

If  we  make  x  =  0,  the  f(x  +  y)  becomes  f(y),  and 
each  of  the  differential  coefficients  being  thus  made  inde 
pendent  of  x,  will  depend  only  on  the  constants  whict 
enter  into  f(x  -H  y\  and  which  also  enter  into  f(y) 
Hence,  if  we  designate  by 

U,  IF,  U",  V",  V"",  &c., 


DIFFERENTIAL    CALCULUS.  53 

the   values   which    the    coefficients    assume    under    this 
hypothesis,  we  shall  have 

f(y)=  U+  Uy+  U"  J^  +  V"^~*r  ^"'^-J 

50.  If  we  take  a  function  of  the  form 

u  =  (a  +  x)n, 
we  shall  have 


&c.  =  &c. 
which  become,  when  we  make   x  =  0, 

U=an,     Uf  =  nan~\     U"  =  n(n-  I)an~*t     &c.; 
hence, 


51.  Remark  1.  The  theorem  of  Maclaurin  has  been 
demonstrated  under  the  supposition  that  the  f(x)  reduces 
to  a  finite  quantity  when  we  make  x  —  0.  The  case, 
therefore,  is  excluded  in  which  x  =  0  renders  the  function 
infinite.  Thus,  if  we  have 

u  =  cot  x,    u~  cosec  a?,     or    u  =  log  x, 

and  make   x  =  0,  we  find  u  =  oo  ;  hence,  neither  of  these 
functions  can  be  developed  by  the  theorem  of  Maclaurin. 


54  ELEMENTS    OF    THE 

Remark  2.  We  have  already  seen  (Art.  45.),  that  the 
theorem  of  Taylor  does  not  apply  to  those  cases  in 
which  the  form  of  the  function  is  changed  by  attributing 
a  particular  value  to  one  of  the  variables  :  the  theorem 
therefore  fails  for  particular  values,  but  is  true  for  all 
others,  and  hence,  the  general  development  never  fails. 

In  the  theorem  of  Maclaurin  the  failure  arises  from  the 
form  of  the  function  :  hence,  it  is  the  general  development 
which  fails,  and  with  it,  all  the  particular  cases. 

EXAMPLES. 

1  .  Develop  into  a  series  the  function 


a 
2.  Develop  into  a  series  the  function 


u=       a-= 
3.  Develop  into  a  series  the  function 


4.  Develop  into  a  series  the  function 

--(-IT 


DIFFERENTIAL    CALCULUS.  55 


CHAPTER  III. 


Of  Transcendental  Functions. 

52.  If  we  have  an  equation  of  the  form 

u  =  ax, 

in  which  a  is  constant,  it  is  plain  that  u  will  be  a  function 
of  x  ;  and  if  a  be  made  the  base  of  a  system  of  logarithms, 
x  will  be  the  logarithm  of  the  number  u  (Alg.  Art,  257). 
When  the  variable  and  function  are  thus  related  to  each 
ether,  u  is  said  to  be  an  exponential  or  logarithmic  func- 
tion of  x.  (Art.  9). 

53.  The  functions  expressed  by  the  equations 

u  —  sin  x,       u=  cos  x,       u  —  tang  a?,     u  =  cot  x,      &c., 

are  called  circular  functions. 

The  logarithmic  and  circular  functions  are  generally 
called  transcendental  functions,  because  the  relation  be- 
tween the  function  and  variable  is  not  determined  by  the 
ordinary  operations  of  Algebra. 

Differentiation  of  Logarithmic  Functions. 

54.  Let  us  resume  the  function 

/ 

u  =  a*. 


56  ELEMENTS    OF   THE 

If  we  give  to  x  an  increment  A,  we  have 


f 

and  u'-u  =  a**h-a* 

In  order  to  develop  ah,  let  us  make  a  =  l+b,  we  shall 
then  have 


hence, 


from  which  we  see,  that  the  coefficients  of  the  first  power 
of  h  will  be 


replacing  6  by  its  value  o  —  1,  and  passing  to  the  limit, 
we  obtain 


or  if  we  make 


—  —  =  ka*,       or 
ax 


in  which  A;  is  dependent  on  a. 


DIFFERENTIAL    CALCULUS.  57 

The  successive  differential  coefficients  are  readily  found. 

For  we  have 

da* 


ax 


hence,  = 

dor 

(Pa* 


&c. 

drtf 

dxn 


—  a*kn. 


5,5.  It  is  now  proposed  to  find  the  relation  which  exists 
between  a  and  k.  For  this  purpose,  let  us  employ  the 
formula  of  Maclaurin, 

u  =  f(x)  —  U+  U—  +  U" h  C/7"— ~ h  &c. 

If  in  the  function 

u  =  a*, 

and   the  successive  differential  coefficients  before  found, 
we  make  x  =  0,  we  have 

hence, 

if  we  now  make  x  =  -r- ,  we  shall  have 
k 


58  ELEMENTS    OF    THE 

designating  by  e  the  second  member  of  the  equation,  and 
employing  twelve  terms  of  the  series,  we  shall  find 


t 

hence,  ak=e,       therefore      a-=e*. 

But,  2.7182818  is  the  base  of  the  Naperian  system  of 
logarithms  (Alg.  Art.  272)  ;  hence,  the  constant  quantity 
k  is  the  Naperian  logarithm  of  a. 

By  resuming  the  result  obtained  in  Art.  54, 

da*  =  a*k  dx, 

we  see  that  the  differential  of  a  quantity  obtained  by 
raising  a  constant  to  a  power  denoted  by  a  ^variable  ex- 
ponent, is  equal  to  the  quantity  itself  into  the  Naperian 
logarithm  of  the  constant)  into  the  differential  of  the 
exponent. 

56.  If  now  we  take  the  logarithms,  in  any  system,  of 
both  members  of  the  equation 

«•  =  «, 

we  shall  have 


kle  =  la,     or    k  = 

Le 


whence, 


or  by  recollecting  that 
we  have 


du  _  la 
"       ; 


DIFFERENTIAL    CALCULUS.  59 

or,  if  we  regard  x   as  the  function,  and  u  as  the  variable, 
we  have  (Art.  34), 

dx_le_  1 
du      la  a*' 

Let  us  now  suppose  a  to  be  the  base  of  a  system  of 
logarithms.  We  shall  then  have  x=  the  logarithm  of 
u,  la  =  I,  and  le=  the  modulus  of  the  system  (Alg. 
Art.  272)  ;  and  the  equation  will  become 


that  is,  the  differential  of  the  logarithm  of  a  quantity  is 
equal  to  the  modulus  of  the  system  into  the  differential  of 
the  quantity  divided  by  the  quantity  itself. 

57.  If  we  suppose  a  =  e  the  base  of  the  Naperian 
system,  and  employ  the  usual  characteristic  lr  to  desig- 
nate the  Naperian  logarithm,  we  shall  have 


that  is,  the  differential  oj  the  Naperian  logarithm  of  a 
quantity  is  equal  to  the  differential  of  the  quantity  divided 
by  the  quantity  itself. 

The  last  property  might  have  been  deduced  from  the 
preceding  article  by  observing  that  the  modulus  of  the 
Naperian  system  is  equal  to  unity. 

58.  The  theorem  of  Maclaurin  affords  an  easy  method 
of  finding  a  logarithmic  series  from  which  a  table  of 
logarithms  may  be  computed.  If  we  have  a  function  of 
the  form, 


ELEMENTS    OF    THE 


we  have   already  seen  that  the  development  cannot  be 
made,  since  f(x)  becomes  infinite  when  07  =  0  (Art.  51.) 
But  if  we  make 


the  function  will  not  become  infinite  when  x  =  0  ;  and 
hence  the  development  may  be  made. 
The  theorem  of  Maclaurin  gives 


«  =  /(*)=  U+  U>^-+  U"        +  U'"3+  &c- 

If  we  designate  the  modulus  of  the  system  of  the  loga- 
rithms by   A,  we  shall  have 


S 


If  we  now  make   x  —  0,  we  have 

U=0,     Uf  =  A,     U"=-A,     Um  =  2A,  &c.; 
hence, 


This  series  is  not  sufficiently  converging,  except  in 
the  case  when  a?  is  a  very  small  fraction.  To  render  the 
series  more  converging,  substitute  —  x  for  x  :  we  then  have 


DIFFERENTIAL    CALCULUS.  61 

and  by  subtracting  the  last  series  from  the  first,  we  obtain 

;(1+*)-/(i-*)  =  ;(g)=2A(!44+  &c.) 


If  we  make 


1+07  Z  i  Z 

=  1  H  --  ,      we  have      x  = 


1  —  x  n 

and  by  observing  that 


I 

we  have 


from  which  we  can  find  the  logarithm  of  n  +  z  when  the 
logarithm  of  n  is  known.  This  series  is  similar  to  that 
found  in  Algebra,  Art.  270. 

If  we  make  n  =  1,  and  z  —  1,  we  have  ll  =  0,  and 


If  we  make  the  modulus  A  =  1 ,  the  logarithm  will  be 
taken  in  the  Naperian  system,  and  we  shall  have 

I'  2  =  0.693147180, 
2  Lr  2  =  lf  4  =  1.386294360; 

and  by  making  z  =  4,   and  n  =  1 ,   we  have 

V  5  =1.60943791 3, 
and      J'2.+  V  5  =  Z'10  ='2.302585093. 


62  ELEMENTS    OP    THE 

If  we  now  suppose  the  first  logarithms  to  have  been 
taken  in  the  common  system,  of  which  the  base  is  10,  we 
shall  have,  by  recollecting,  that  the  logarithms  of  the  same 
number  taken  in  two  different  systems  are  to  each  other 
as  their  moduli  (Alg.  Art.  267), 

Z10     :     no     :  :     A     :     1, 
or,  1     :     2.302585093     :  :     A     :     1 ; 

WhenC6'  A=  2.30258509  =°-434284488- 

Remark.  To  avoid  the  inconvenience  of  writing  the 
modulus  at  each  differentiation  (Art.  56),  the  Naperian 
logarithms  are  generally  used  in  the  calculus,  and  when 
we  wish  to  pass  to  the  common  system,  we  have  merely 
to  multiply  by  the  modulus  of  the  common  system.  We 
may  then  omit  the  accent,  and  designate  the  Naperian 
logarithm  by  Z. 

59.  Let  us  now  apply  these  principles  in  differentiating 
logarithmic  functions. 

1.  Let  us  take  the  function        u  =  l 
Make 
and  we  shall  have 


but 


DIFFERENTIAL    CALCULUS. 


63 


whence, 


2.  Take  the  function    u  = 

ar-  Vl-x 

and  make    i/l+  x+  -\/l—x=y,     -\/\-\-x—  *\/l—x=z, 
which  gives 


tl==/m=Zy_fe>       and 


y       z 


But  we  have 
dx 


,  dx  dx  —  dx     /    /-—  -      /^   —  \ 

dy=  —  -  --  :  --  7=  =  —  7  --  ~  (  V1  +x~  V  1  —  g)> 
2Vl  +  a?     2V1  —  a;     2V1  —  a^v  / 


d  —      ^ 


zdx 

T 

dx  dx 


T+x+  */l^x), 


Whence, 


dz 


zdx 


ydx 


and  observing  that  yz+zz=4:  and  yz  = 

dx 


we  have 


du  =  - 


xVl- 


64 


ELEMENTS    OF    THE 


(fe 


4'   w=: 


5.     w  = 


7 
6.     u  —  l 


, 
a7—  Va  —  ad 


60.  Let  us  suppose  that  we  have  a  function  of  the  form 


Make  Ix  —  z,  and  we  have 

u  —  zn,  du  =  nz*~ldz, 

and  substituting  for  z  and  dz  their  values, 


a? 


61.  Let  us  suppose  that  we  have 


Make  /a?  =  z,  and  we  shall  have, 


hence, 


7  j       dz  ,       dx 

u  =  lz9        du  =  — ,         dz  —  — : 
z  x 

dx 
U~    xlx 


DIFFERENTIAL    CALCULUS.  65 

62.  The  rules  for  the  differentiation  of  logarithmic  func- 
tions are  advantageously  applied  in  the  differentiation  of 
complicated  exponential  functions. 

1.  Let  us  suppose  that  we  have  a  function  of  the  form 

«  =  *», 

in  which  z  and  y  are  both  variables. 

If  we  take  the  logarithms  of  both  members,  we  have 

lu  =  ylz  ; 

du      j  ,          dz 
hence,  —  =  dylz  +  y  —  ; 

or,  du  =  ulzdy  +  uy  — , 

or  by  substituting  for  u  its  value 

du  =  dzy  —  zylzdy  -j-  yzy~ldz. 

Hence,  the  differential  of  a  function  which  is  equal  to 
a  variable  root  raised  to  a  power  denoted  by  a  variable 
exponent,  is  equal  to  the  sum  of  the  differentials  which 
arise,  by  differentiating,  first  under  the  supposition 
that  the  root  remains  constant,  and  then  under  the  sup 
position  that  the  exponent  remains  constant  (Arts.  55, 
and  32). 

2.  Let  the  function  be  of  the  form 

u=a»*. 
Make,    b*  =  y,    and  we  shall  then  have  (Art.  55), 

u  =  ay,         du  —  aylady  ;     but    dy  =  b*lbdx, 

hence,  du  =  afb'lalbdx. 

5 


66  ELEMENTS    OP   THE 

3.  Let  us  take  as  a  last  example 
u  =  z<, 

in  which  z,  t,  and  s,  are  variables. 
Make,      t'  =  y,    we  shall  then  have 

u  =  zy,          dw  =  zy  Izdy  +  yzy~l  dz. 
But         dy  =  fltds  +  stf-  W*  ; 
hence,        cfo  =  zflz(fltds  +  **-'cfr)  +  Vaf~ldz, 


Differentiation  of  Circular  Functions, 

63.  Let  us  first  find  the  differential  of  the  sine  of  an 
arc.  For  this  purpose  we  will  assume  the  formulas  (Trig. 
Art  XIX), 

sin  a  cos  b  -f  sin  b  cos  a 


sin  (a  +  b)  = 
sin  (a  —  b)  = 


s 

sin  a  cos  6  —  sin  b  cos  a 


R 

If  we  subtract  the  second  equation  from  the  first, 

.    ,        ,x       .    ,        ,N      2  sin  6  cos  a 
sin  (a  +  6)  —  sin  (a  —  b)  = — . 

and  if  we  make  a  -f  b  =  x  +  h,  and  a  —  b  =  a?,   we  shall 
have 

2sin  — Acos  (x  H 

sin  (x  +  h)-—  sino?  = 


DIFFERENTIAL    CALCULUS.  67 

and  dividing  both  members  by  A, 

2sin  —  h  cos  (x-\  --  h] 
2  \         2    / 


—  sma? 


hR 

sin  —  h     cosfoH  --  h\ 


If  we  now  pass  to  the  limit,  the  second  factor  of  the 

r»r\c   *Y» 

second  member  of  the  equation  will  become     —  —  . 

R 

sin  —  h 

2 
In  relation  to  the  first  factor  —  j-  -  its  limit  will  be  unity. 

¥* 

-r,  Rsina  sina       cosa 

ror,        tang  a  =  -  ,     whence     -  =          ; 
cosa  tanga        R    ' 

Now,  since  an  arc  is  greater  than  its  sine  and  less  than 
its  tangent* 

sina     -  ,      sina       sina 

and     -  > 


a  a        tanga 


*  The  arc  DB  is  greater  than  a  straight  line 
drawn  from  D  to  B,  and  consequently  greater 
than  the  sine  DE  drawn  perpendicular  to  JIB. 

The  area  of  the  sector  J1BD  is  equal  to 
-  JIB  X  BD,  and  the  area  of  the  triangle  J2BC 

is  equal  to  -JIB  X  BC.    But  the  sector  is  less        A  E   B 

than  the  triangle  being  contained  within  it  :  hence, 


consequently,  BD  <    BC. 


00  ELEMENTS    OF    THE 

hence,  the  ratio  of  the  sine  divided  by  the  arc  is  nearer 
unity  than  that  of  the  sine  divided  by  the  tangent.  But 
when  we  pass  to  the  limit,  by  making  the  arc  equal  to  0, 
the  sine  divided  by  the  tangent  being  equal  to  the  cosine 
divided  by  the  radius,  is  equal  to  unity  :  hence  the  limit 
of  the  ratio  of  the  sine  and  arc,  is  unity. 

When  therefore  we  pass  to  the  limit  by  making  h  =  0, 
we  find 

d  sina?  _  cos  a?  m 
~dx~     ~~R~ 

cosxdx 


,  ,  . 

hence,  a  sm  x  = 


— 
K 


64.  Having  found  the  differential  of  the  sine,  the  diffe- 
rentials of  the  other  functions  of  the  arc  are  readily  de- 
duced from  it. 

cosa?  =  sin(90°  —  a?),         dcosx  =  dsin(90°  —  a?), 
and  by  the  last  article, 

dsin(90°  -  x)  =  -^cos(90°  -  x)d(QQ°  -  a?), 
K 

=  —  -rrcos  (90°  —  x)dx : 
K 


7  sin  xdx 

hence,  d  cos  x  = = —  ; 

K 


the  differential  of  the  cosine  in  terms  of  the  arc  beii  , 
negative,  as  it  should  be,  since  the  cosine  and  arc  axe 
decreasing  functions  of  each  other  (Art.  31.) 


DIFFERENTIAL    CALCULUS.  69 

65.  Since  ihe  versed  sine  of  an  arc  is  equal  to  radius 
minus  the  cosine,  we  have 


d  T  er-sin  x  =  d  (R  -  cos  o?)  = 


. 
R 


66.  Since    tang  x  =  R  Sm  x  ,    we  have  (Art.  30), 

COS  X 


7f  R  cos  x  d  sin  x  —  R  sin  x  d  cos  x 

a  wng  A  =  -  _  - 

2 


C080? 

(cos2o?4-  sin2o? 


but  cos2o?  +  sin2o?  =  R2  : 

hence,  d  tango?  =         , 

cos^o? 


67.   Since          cota-  = ,         we  have 

tango? 


tang2o?  tang2^  cos2o?  ' 

but,  tang2a; 

R2dx 


COS0? 


hence,  ^  cot  a?  = 


— 
sm2o? 


which  is  negative,  as  it  should  be,  since  the  cotangent  is  a 
decreasing  function  of  the  arc. 


70  ELEMENTS    OF   THE 

R2 

68.     Since      seca?  =  -  ,         we  have 
cos  x 

,  R2d  cosx        R  s'mxdx 

dsecx  =  --  -  -  =  -  -  -  : 
cos  a?  cos  a? 

R  sin  x  R2 

but,         -  =  tango?,       and      -  =  seca?; 
cos  a?  cos  a? 


7  seca? 

hence,  a  sec  a?  = 


. 

R2 

69.     Since     cosec  a?  =  —  -  -  ,         we  have 
sin  a? 

,  R2d  sin  a?  R  cosxdx 

a  cosec  a?  =  --  —  -  =  --  —  -  : 


coseca?  cotxdx     > 
hence,  d  cosec  x  = 


70.  If  we  make     R  =  l,     Arts.  63,  64,  65,  66,  67, 
will  give, 


d  sina?=  cos  a?  da? 

(i), 

dcosx=  —  s'mxdx 

(2), 

d  ver  sina?  =  sina?da? 

(3), 

dx 

(4), 
CM. 

cosaa? 

dx 
a  cot.  .7?  —  

BUT* 

The  differential  values  of  the  secant  and  cosecant  are 
omitted,  being  of  little  practical  use. 

71.  In  treating  the  circular  functions,  it  is  found  to  be 
most  convenient  to  regard  the  arc  as  the  function,  and  the 


DIFFERENTIAL    CALCULUS.  71 

sine,  cosine,  versed-sine,  tangent,  or  cotangent,  as  the 
variable.  If  we  designate  the  variable  by  u,  we  shall 
have  in  (Art.  63)  sin  x  =  u,  and 

Rdu  Rdu 


cosa? 
If  we  make    coso?  =  w,    we  have  (Art.  64), 

,  Rdu  Rdu 

sin  x 

If  we  make    ver-sina?  =  u,    we  have  (Art.  65), 

Rdu 
dx  =  — 


smx 


But,          sina?=  V-R2— cos2#,     and     coso?=.R— tt, 
therefore,  cos2a?  =  R2  —  2  Ru  +  u2, 


hence,  sin  a?  =  -\/2Ru  —  u2, 

''• 

,  Rdu 

and  consequently,         ax  =     ,  —  m 

V2Ru  —  u2> 
If  we  make   tang  x  =  u,    we  have  (Art.  66) 


cos^r  R  cos2x       R2  R2 

but    —  =—  =  -  ,     hence 


R     ~sec*'  R2  ~sec2o?~fl2+tangV 

R2du 
hence,  -  dx  = 


72  ELEMENTS    OF    THE 

Now,  if  we  make      R  =  l,      the   four   last  formulas 
become 


du 

( 

i~                du 

« 

7P3T' 

Vl  -  u2  ' 

rfr- 

du 

h     Jw 

and  these  formulas  being  of  frequent  use,  should  be  care- 
fully committed  to  memory. 

72.  The  following  notation  has  recently  been  introduced 
into  the  differential  calculus,  and  it  enables  us  to  designate 
an  arc  by  means  of  its  functions. 

sin~xw  =  the  arc  of  which  u  is  the  sine, 
cos"1^^  the  arc  of  which  u  is  the  cosine, 

tang-1M  —  the  arc  of  which  u  is  the  tangent, 
&c.  &c.  &c. 

If,  for  example,  we  have 

du 


=  sin  X       then,       dx  = 


1-w2 


73.  We  shall  now  add  a  few  examples. 
1  .  Let  us  take  a  function  of  the  form 


Make  cos  x  —  z^     and 

then,  u  =  zy,    and     (Art.  62); 

du  =  zy  Izdy  -h  yzy~ldz: 


DIFFERENTIAL    CALCULUS.  73 

also,  dz=.  —  smxdx,     and     dy  =  cosxdx 

hence,  du  =  zy  (lz  dy  +  —  dz  \ 

=  cosa?rinrUcosa?cosa?  --  }dx. 
\  cosa?  / 

2.  Differentiate  the  function 

mdu 

x  =  sm     mu.  ax  =  —  , 

Vl-mV 

3.  Differentiate  the  function 

x  =  cos"1  (u  Vl  —  u2J 


4.  Differentiate  the  function 

w  ,          2du 


5    Differentiate  the  function 

x  =  sin"1  (%u  Vl  —  w2  ), 

v  / 

6.  Differentiate  the  function 


_!  a:  .,        ydx  —  xdy 

l—  ,  du  =  — 

y 


74.  We  are  enabled  by  means  of  Maclaurin's  theorem 
and  the  differentials  of  the  circular  functions,  to  find  the 


74  ELEMENTS    OF    THE 

value  of  the  principal  functions  of  an  arc  in  teims  of  the 
arc  itself. 

Let  u=  f(x)  =  sin  a?:     then, 

du  tPu  d?u 


,  —  > 

dx  da?  dx3 


=  —  COSX> 


-j-r  =  +  COS  X. 

x  dx5 

If  we  now  render  the  differential  coefficients  independent 
of  x,  by  making  x  =  0,  we  have  (Art.  49), 


X  0?  X5 

:-+ 


75.  To  develop  the  cosine  in  terms  of  the  arc,  make 

u  =  f(x)  =  cos  x  ;     then, 

du  .  d2u  d?u 

—  =—  sma?,         -—-  —  —  cosa?,         -Y-^-^sma?, 

c^a?  c^r2  dx3 

d*u  d5u 

-—  =  cosx,         -7-j  =  —  sm  a:, 

dx*  dx5 

and  rendering  the  coefficients  independent  of  a?,  we  have 
17=1,         ^=0,         V"=-l,         U"'=0, 


hence,          cosx=  1  --        +  -  &c. 


DIFFERENTIAL    CALCULUS.  75 

The  last  two  formulas  are  very  convenient  in  calculating 
the  trigonometrical  tables,  and  when  the  arc  is  small  the 
series  will  converge  rapidly.  Having  found  the  sine  and 
cosine,  the  other  functions  of  the  arc  may  readily  be 
calculated  from  them. 

76.  In  the  two  last  series  we  have  found  the  values  of 
the  functions,  sine  and  cosine,  in  terms  of  the  arc.     We 
may,  if  we  please,  find  the  value  of  the  arc  in  terms  of 
any  of  its  functions. 

77.  The  differential  coefficient  of  the  arc  in  terms  of 
its  sine,  is  (Art.  71), 

^_       1  2x4 

^-7T^-(1" 

developing  by  the  binomial  formula,  we  find 
dx  1  l-3         1.3.5 


In  passing  from  the  function  to  the  differential  coeffi- 
cient, the  exponent  of  the  variable  in  each  term  which 
contains  it,  is  diminished  by  unity  ;  and  hence,  the  series 
which  expresses  the  value  of  x  in  terms  of  u,  will  contain 
the  uneven  powers  of  u,  or  will  be  of  the  form 


Cu5  +  Du7  +  &c.; 
and  the  differential  coefficient  is 

^  =  A 
au 


76  ELEMENTS    OF    THE 

But  since  the  differential  coefficients  are  equal  to  each 
other,  we  find,  by  comparing  the  series, 


~2.3'         *CO'         -O^T7' 

hence, 

u       I   u3     l.3u5        1.3.5      7 

^p— —  gin — ^T/ — U  -\~   Q£C 

1       2    3      2.4.5     2.4.6.7 

If  we  take  the  arc  of  30°,   of  which  the  sine  is    — 
(Trig.  Art.  XV),  we  shall  have 


and  by  multiplying  both  members  of  the  equation  by  6, 
we  obtain  the  length  of  the  semi-circumference  to  the 
radius  unity. 

78.  To  express  the  arc  in  terms  of  its  tangent,  we  have 
(Art.  71), 


which  gives 

/Jv 
^±^1_ 

du 
hence  the  function  x  must  be  of  the  form 

ac  =  Au  +  Bit3 

and  consequently 


=  A 
du 


DIFFERENTIAL    CALCULUS.  77 

and  by  comparing  the  series,  and  substituting  for  A,  B,  C, 
&c.,  their  values,  we  find 

u       u?      u5      u1  , 
tf^tang    u  =  —  -  —  +  -j-  —  +&c. 

If  we  make  x  =  45°,  u  will  be  equal  to  1 ;  hence, 

arc  45°  =  1  —  —  + +  &c. 

357 

But  this  series  is  not  sufficiently  convergent  to  be  used 
for  computing  the  value  of  the  arc.  To  find  the  value 
of  the  arc  in  a  more  converging  series,  we  employ  the 
following  property  of  two  arcs,  viz. : 

Four  times  the  arc  whose  tangent  is  — ,  exceeds  the 

5 

arc  of  45°  by  the  arc  whose  tangent  is    — —  *. 

239 


*  Let  a  represent  the  arc  whose  tangent  is    — .      Then  (Trig.  Art. 

XXVI), 

2  tang  a          5 


2  tans;  2  a    __  120 
~  1—  tang22  a  =   119  ' 

The  last  number  being  greater  than  unity,  shows  that  the  arc    4  a    ex- 
ceeds 45°.    Making 


the  difference,    4  a  —  45°  ==  A  —  B  =  6,    will  have  for  its  tangent 


hence,  four  times  the  arc  whose  tangent  is  -—,  exceeds  the  arc  of  45°  by  an 

arc  whose  tangent  is    -  . 
/          239 


78 
But 

tang- 
hence, 


ELEMENTS    OF    THE 


-iJL_.l      i       i       i 

ang     5  ~  5        S.S^S.S5      7.57 


239      239      3(239)3      5(239)5      7(239)7 


arc  45°  = 


\239      3(239)3      5(239)5      7(239) 

Multiplying  by  4,  we  find  the  semi-circumference 
=  3.141592653. 


| 
7 


DIFFERENTIAL    CALCULUS.  79 


CHAPTER  IV. 


Development  of  any  Function  of  two  Variables 
— Differential  of  a  Function  of  any  number 
of  Variables — Implicit  Functions — Differential 
Equations  of  Curves — Of  Vanishing  Fractions. 

79.  We  have  explained  in  Taylor's  theorem  the  method 
of  developing  into  a  series  any  function  of  the  sum  or  dif- 
ference of  two  variables. 

We  now  propose  to  give  a  general  theorem  of  which 
that  is  a  particular  case,  viz  : 

To  develop  into  a  series  any  function  of  two  or  more 
variables,  when  each  shall  have  received  an  increment) 
and  to  find  the  differential  of  the  function. 

80.  Before  making  the  development  it  will  be  necessary 
to  explain  a  notation  which  has  not  yet  been  used. 

If  we  have  a  function  of  two  variables,  as 

u  =/(*>  y), 

we  may  suppose  one  to  remain  constant,  and  differentiate 
the  function  with  respect  to  the  other. 

Thus,  if  we  suppose  y  to  remain  constant,  and  x  to 
vary,  the  differential  coefficient  will  be 

•-/(*  y);    (i), 


80  ELEMENTS    OF    THE 

and  if  we  suppose  x  to  remain  constant  and  y  to  vary, 
the  differential  coefficient  will  be 


The  differential  coefficients  which  are  obtained  under 
these  suppositions,  are  called  partial  differential  coef- 
ficients. The  first  is  the  partial  differential  coefficient 
with  respect  to  x,  and  the  second  with  respect  to  y. 

81.  If  we  multiply  both  members  of  equation  (1)  by 
dx,  and  both  members  of  equation  (2)  by  dy,  we  obtain 

d-^dx=f'(x,y)dx,     and     '-dy  =  f"(x,y)dy. 

The  expressions, 

du  ,  du  j 

—dx,  —dy, 

dx  ay   " 

are  called  'partial  differentials;  the  first  a  partial  diffe- 
rential with  respect  to  x,  and  the  second  a  partial  diffe- 
rential with  respect  to  y  :  hence, 

A  partial  differential  coefficient  is  the  differential  co- 
efficient of  a  function  of  two  or  more  variables,  under 
the  supposition  that  only  one  of  them  has  changed  its 
value  :  and, 

A  partial  differential  is  the  differential  of  a  function 
of  two  or  more  variables,  under  the  supposition  that  only 
one  of  them  has  changed  its  value. 

82.  If  we  differentiate  equation  (1)   under  the  suppo- 
sition that  x  remains  constant  and  y  varies,  we  shall  have 


dy 


DIFFERENTIAL    CALCULUS.  81 

and  since  x  and  dx  are  constant 

_d(du) 
~ 


which  we  designate  by 

d*u 


d2u 
hence' 


The  first  member  of  this  equation  expresses  that  the 
function  u  has  been  differentiated  twice,  once  with  respect 
to  x,  and  once  with  respect  to  y. 

If  we  differentiate  again,  regarding  x  as  the  variable, 
we  obtain 

dzu         riv,      N 

__=/>>,y), 

which  expresses  that  the  function  has  been  differentiated 
twice  with  respect  to  x  and  once  with  respect  to  y.  And 
generally 

dn*mu 

indicates  that  the  function  u  has  been  differentiated  n  -f  m 
times,  n  times  with  respect  to  a?,  and  m  times  with  respect 
to  y. 

83.  Resuming  the  function 


if  we  suppose  y  to  remain  constant,  and  give  to  x  an  arbi- 
trary increment  h,  we  shall  have  from  the  theorem  of  Taylor, 

duh    ,  d2u   h2    ,  d3u     h3 
/(*  +  **)  =  «  +  _+^  —  +  -^~+&c., 


82  ELEMENTS    OF    THE 

,  .  ,  du          d2u          d3u 

mwh,ch,  u,     -,  , 


are  functions  of  x  and  y,  and  dependent  on  the  constants 
which  enter  the  f(x,y). 

If  we  now  attribute  to  y  an  increment  k,  the  function 
ut  which  depends  on  y,  will  become 

d2u  kz       d'3u     k3 


and  the  function  —  will  become 
dx 

du       ffu     k        d3u      k2         d*u       k3  „ 


dx     dxdy  1       dxdy2  l.2dxdy3  1.2.3 

and  the  function     -y-r-,     will  become 
dor 


d?u    k         d*u     k2         d5u        k3         & 

1.2.3  " 


and  the  function    -  will  become 


d*u     k          d5u       k2          d?u        A3          & 
1  "  da?dy*   1.2       ^rfy3  1.2.3 


&c.  &c.  &c.  &c. 

Substituting  these  values  in  the  development  of 

/(*  +  *,  y), 


DIFFERENTIAL    CALCULUS.  83 

and  arranging  the  terms,  we  have 

duk       d?u   k2      tfu    k3 


du  h       a?u   hk       (Pu     htf          „ 


ffu    tf_      cPu    tfk_ 
V~+  + 


which  is  the  general  development  of  a  function  of  two 
variables,  when  each  has  received  an  increment,  in  terms 
of  the  increments  and  differential  coefficients. 

84.  If  we  transpose  u  =f(x,  y)  into  the  first  member, 
and  apply  the  result  of  Art.  19  to  a  function  of  two  vari 
ables,  we  find 


y.  ,  =.        ,  ;il 

The  differential  of  f(x,  y)  =  du,  which  is  obtained  under 
the  supposition  that  both  the  variables  have  changed  their 
values,  is  called  the  total  differential  of  the  function. 

85.  If  we  have  a  function  of  three  variables,  as 
u  =  f(x,  y,  z), 

and  suppose  one  of  them,  as  z,  to  remain  constant,  and 
increments  h  and  k  to  be  attributed  to  the  other  two,  the 
development  of  /  (x  +  h,  y  +  k,  z)  will  be  of  the  same 
form  as  the  development  of  f(x-\-h,  y  +  k)  ;  but  u  and 
all  the  differential  coefficients  will  be  functions  of  z 


84  ELEMENTS    OF    THE 

If  then  an  increment  I  be  attributed  to  z,  there  will  be 
four  terms  of  the  development  of  the  form 

du  ,          du  ,          du  7 
u>        T*1,        -T-/C,         -j-l. 
ax  ay  dz 

If  u  were  a  function  of  four  variables,  as 

«  =  /(*»  y>  *>  *)> 

there  would  be  five  terms  of  the  form 

du,          du,          du1         du 
u,        T~">        T-ft*        T~*>        -=-»i  ; 
cfo  dy  dz  <fo 

and  a  new  variable  introduced  into  the  function,  would 
introduce  a  term  containing  the  first  power  of  its  increment 
into  the  development. 

If  we  transpose  u  into  the  first  member,  and  make  the 
same  supposition  as  in  the  last  article,  we  shall  have 

du          du          du 
y,  z)-]  =  -dx  +  -dy  +  -dz, 


and,  for  like  reasons, 

.,.    _    .,  N_      du          du          du          du 

d\J(x,  y,  z,  s)]  =  -fa&x  +  jydy  +  -j~zdz  +  -j^ds, 

from  which  we  may  conclude  that,  the  total  differential 
of  a  function  of  any  number  of  variables  is  equal  to  the 
sum  of  the  partial  differentials. 

86.  The  rule  demonstrated  in  the  last  article  is  alone 
sufficient  for  the  differentiation  of  every  algebraic  function 
1.  Let         u  =  oc?  +  y3  —  z;     then 


-—  (Lx  =  2  x  dx.       1st  partial  differential  ; 
ax 


DIFFERENTIAL    CALCULUS, 

j-dy  =  3y2dy,     2d  partial  differential  ; 

~dz=—dz,        3d         "  « 

hence,  du  =  2xdx  +  3y2dy  —  dz. 

2.  Let    u  =  xy;    then, 


rfll 


du  =  ydx  +  x  dy. 
3.  Let         u  =  xmyn;       then, 

C?M  ,         • 

--.  cte  =  mxm~lyndx, 

du 

•—dy  =  nyn~lxmdy  :        hence, 


nxdy\ 


4.  Let         w  =  ~;         then, 

du  ,  dx 
---dx  =  —, 
dx  y 

du 


hence, 


86  ELEMENTS    OP    THE 

5.  Let        u  =  -  /-^—=  =  ay(x2  +  y2)"T ;     then, 
du  ,  ayxdx 

n    _r    rj  /yt   —     *s 

*;*,=    adv       a?dy . 


,  , 

hence,  du  = 

6.  Let        u  —  xyzt;      then, 

du  =  yztdx+xztdy  +  xytdz  +  xyzdt. 

7.  Let         u  =  zv;         then, 

> 

(Art.  55), 


dz      (Art.  32) 
dx 

hence,  du  —  &lzdy  +  yzy~ldz. 

Remark.  In  chapter  II,  the  functions  were  supposed 
to  depend  on  a  common  variable,  and  the  differentials  were 
obtained  under  this  supposition.  We  now  see  that  the  dif 
ferentials  are  obtained  in  the  same  manner,  when  the  func- 
tions are  independent  of  each  other,  and  unconnected  with 
a  common  variable. 

87.  We  have  seen  (Art.  39),  that  a  function  of  a  single 
variable  has  but  one  differential  coefficient  of  the  first 
order,  one  of  the  second,  one  of  the  third,  &c.  ;  while  9 


DIFFERENTIAL    CALCULUS.  87 

function  of  two  variables  has  two  differential  coefficients 
of  the  first  order,  a  function  of  three  variables,  three  ;  a 
function  of  four  variables,  four  ;  &c. 

It  is  now  proposed  to  find  the  successive  differentials 
of  a  function  of  two  variables,  and  also  the  successive 
differential  coefficients. 

We  have  already  found 

,        du  ,     ,  du  , 

du  =  —-  ax  +  —  ay. 
ax          ay  " 

du          du 
But 


du  du 

and  since,  -j-    and  -y-    are  functions  of   x    and    y,   the 

dx  dy 

differentials    -j-<ir,  ~rdy,    must  each   be   differentiated 

dx  dy  yi 

with  respect  to  both  of  the  variables  ;  dx  and  dy  being 

supposed  constant  :  hence, 

/du  d2u                d2u 


/du     \       d2u  d2u 

and  d(d 

hence  we  have 


dxdy 
If  we  differentiate  again,  we  have 


88  ELEMENTS    OF    THE 


and  consequently, 


It  is  very  easy  to  find  the  subsequent  differentials,  by 
observing  the  analogy  between  the  partial  differentials  and 
the  terms  of  the  development  of  a  binomial. 

We  also  see  that,  a  function  of  two  variables  has  two 
partial  differential  coefficients  of  the  first  order,  three  of 
the  second,  four  of  the  third,  &c. 

88.  There  are  several  important  results  which  may  be 
deduced  from  the  general  development  of  the  function  of 
two  variables  (Art.  83). 

1st.  If  we  make  a?  =  0,  and  y  =  0,  u  and  each  of 
the  differential  coefficients  will  become  constant,  and  we 
shall  have 

i.,    .  du 


d2u  ,2          d2u    ,  7       cw  72\ 
r-T5-"  +  2,    7    hk  +  -rzk 2) 
1.2\da?  dxdy  dy2     ) 

+   &c., 

which  is  the  development  of  any  function  of  two  variables 
in  terms  of  their  ascending  powers,  and  coefficients  which 
are  dependent  on  the  constants  that  enter  the  primitive 
function. 

2d.  If,  in  the  general  development,  we  make  y  =  Q,  and 
k  =  0,  we  shall  have 


DIFFERENTIAL    CALCULUS. 

du  h      d*u    h2        d*u 


T      ___      _T_ 

which  is  the  theorem  of  Taylor. 

3d.  If  we  make  y  —  0,  k  =  0,  and  a?  =  0,  we  have 


or,     /(  h)  =  U  +  U'h  +  17"-^  +  ^7^73+,  &c.  ; 
which  is  the  theorum  of  Maclaurin. 

Implicit  Functions. 

89.  When    the   relation    between   a   function   and   its 
variable  is  expressed  by  an  equation  of  the  form 


in  which  y  is  entirely  disengaged  from  x,  y  has  been 
called  an  explicit  ,  or  eocpressed  function  of  #  (Art.  6). 
When  y  and  x  are  connected  together  by  an  equation  of 

the  form 

/(*,?)  =  0, 

y  has  been  called  an  implicit,  or  implied  function  of  x 
(Art.  6.) 

It  is  plain,  that  in  every  equation  of  the  form 


y  must  be  a  function  of  07,  and  a?  of  y.  For,  if  the 
equation  were  resolved  with  respect  to  either  of  them,  the 
value  found  would  be  expressed  in  terms  of  the  other 
variable  and  constant  quantities. 


90  ELEMENTS    OF   THE 

90.  If  in  the  equation 

u~  /(ff>y)  =  0» 

we  suppose  the  variables  x  and  y  to  change  their  values 
in  succession,  any  change  either  in  x  or  y,  will  produce  a 
change  in  u  :  hence,  u  is  a  function  of  x  and  y  when 
they  vary  in  succession.  The  value,  however,  which  u 
assumes,  when  a?  or  y  varies,  will  reduce  to  0  when 
such  a  value  is  attributed  to  the  other  variable  as  will 
satisfy  the  equation 

X*,y)  =  o.  ( 

We  have  from  Art.  83, 

f(x  +  h,  y  +  k)  —  u  =  j-j+  terms  containing  A2, 

du 

-jyi  +  terms  containing  A2, 

plus  other  terms  containing  kh,  and  the  higher  powers  of 
h  and  k. 
But,  since  y  is  a  function  of  #,  we  have 


in  which  P  is  the  differential  coefficient  of  y  regarded  as 
a  function  of  x.  Substituting  this  value  of  k,  and  we 
have 

du 
(x  +  h,  y  -f  k)  —  u  =  j-Ph  +  terms  containing  h2, 

du 

-j-h  +  terms  containing  A2, 

plus  other  terms  containing  the  higher  powers  of  h. 


DIFFERENTIAL    CALCULUS.  91 

But,  in  consequence  of  the  relation  between  y  and  a?, 
the  first  member  of  the  equation  will  be  constantly  equal 
to  0.  Hence,  by  the  law  of  indeterminate  coefficients 
(Alg,  Art.  244), 

du         du\ 

p+^=°;  v;;  ?'"',:! 

du 

hence,  P  =  f=-^. 

dx  du 

Ty 

Hence,  the  differential  coefficient  of  y   regarded  as  a 
function  of  x,  is  equal  to  the  ratio  of  the  partial  differen- 
tial coefficients  of  u  regarded  as  a  function  of  x,  and  u 
regarded  as  a  function  ofy,  taken  with  a  contrary  sign. 
Let  us  take,  as  an  example,  the  equation 


du  da 

then'  dx=2x>     and     Ty==<*y: 

du. 

dx  x        dy 

hence'  du  =  --j-  =  te 

dy 

Although  the  differential  coefficient  of  the  first  order  is 
generally  expressed  in  terms  of  x  and  y,  yet  y  may  be 
eliminated  by  means  of  the  equation  f(x,y)  =  0,  and  the 
coefficient  treated  as  a  function  of  x  alone.  In  the  above 
equation  we  have 


92  ELEMENTS    OF    THE 

dy  x 

hence,  —  = / 

dx  VR2-x?      .,. 

92.  If  it   be   required   to   find   the   second   differential 
coefficient,  we  have  merely  to  differentiate  the  first  diffe- 
rential coefficient,  regarded  as  a  function  of  x,  and  divide 
the  result  by  dx.     Thus,  if  we  designate  the  first  diffe- 
rential coefficient  by  p,  the   second  by  g,  the  third  by 
r,  &c.,  we  shall  have 

dp  da 

^JT_    .     —     xy  _i_     __    A*  fVf* 

dx        '  dx 

93.  To  find  the   second  differential  coefficient  in  the 
equation  of  the  circle,  we  have 

dy  _        x 
dx  yf 


,  /dy \  _  —  ydx  -f-  xdy 
\dx)~~       v* 


hence' 


and  by  substituting  for     -^    its  value     —  —  ,     we  have 
dx  y 


1.  Find  the  first   differential  coefficient   of  y,   in   the 
equation 

y2  —  2mxy  -f  x2  —  a2  =  u  =  0, 

du  du 

—  —  —  Zmy  +  2x,  —  =  2y 

dx  dy 


DIFFERENTIAL    CALCULUS. 


hence,  ^  =  -  f 

oa?          L 


—  2  77107  J      y  —  mx 

2.  Find  the   first  differential   coefficient   of  y   in  the 
equation 

y2  +  2xy  +  or2  -  a2  =  0. 

' 


3.  Find  the  first  and  second  differential  coefficients  of  yt 
in  the  equation 


3<zy,  -^  =  3  y2  —  3  ax, 

ax  dy 

hence,  ^y_      3x?-3ay  =  ay-a? 

dx          3y2—3ax     -f  —  ax 

For  the  second  differential  coefficient,  we  have 


or,  by  substituting  for     -j-    its  value,  and  reducing, 
ax 


2xy  (y3  —  3  axy  +  a?3)  +  2a3a?y 
tf-axj* 

but  from  the  given  equation 

y5—  Saxy  +  ^  =  0. 

o^y  2a?xy 

hence,  —^  =  —  —  —  2_  . 

rfaT2          (y2—  axf 


94  ELEMENTS    OF   THE 

Differential  Equations  of  Curves. 

94.  The  Differential  Calculus  enables  us  to  free  an 
equation  of  its  constants,  and  to  find  a  new  equation  which 
shall  only  involve  the  variables  and  their  differentials. 

If,  for  example,  we  take  the  equation  of  a  straight  line 

y  =  ax  +  bt 
and  differentiate  it,  we  find 

dx 
and  by  differentiating  again, 


The  last  equation  is  entirely  independent  of  the  values 
of  a  and  b,  and  hence,  is  equally  applicable  to  every 
straight  line  which  can  be  drawn  in  the  plane  of  the  co- 
ordinate axes.  It  is  called,  the  differential  equation  of 
lines  of  the  first  order. 

95.  If  we  take  the  equation  of  the  circle 


and  differentiate  it,  we  find 

xdx  -\-  ydy  =  0. 

This  equation  is  independent  of  the  value  of  the  radius 
R,  and  hence  it  belongs  equally  to  every  circle  whose 
centre  is  at  the  origin  of  co-ordinates. 


DIFFERENTIAL    CALCULUS.  95 

96.  If  the  origin  of  co-ordinates  be  taken  in  the  circum- 
ference, the  equation  of  the  circle  (An.  Geom.  Bk.  Ill, 
Prop.  I,  Sch.  3)  is 


from  which  we  find 


and  by  differentiating, 


A  _  2xdx)  —  (y2  +  xz)dx 

~~~ 


or  by  reducing 

(x2  -y*)dx  +  2xydy  =  0, 

which  is  the  differential  equation  of  the  circle  when  the 
origin  of  co-ordinates  is  in  the  circumference. 

The  last  equation  may  be  found  in  another  manner. 

If  we  differentiate  the  equation  of  the  circle, 

yz  =  2Rx-x2, 
we  have,  after  dividing  by  2 

ydy  =.  Rdx  —  xdx  ; 


hence, 


ax 


If  this  value  of  R  be  substituted  in  the  equation  of  the 
circle,  we  have 


(a?  — 
the  same  differential  equation  as  found  by  the  first  method. 


96  ELEMENTS    OF    THE 

97.  If  we  take  the  general  equation  of  lines  of  the 
second  order  (An.  Geom.  Bk.  VI.  Prop.  XII,  Sch.  3), 

y2  =  mx  +  nx2, 
and  differentiate  it,  we  find 

2ydy  =  mdx  -f  2nxdx ; 

differentiating  again,  regarding  dx  as  constant,  we  have, 

after  dividing  by  2, 

\ 

dy2  -f-  yd?y  =  ndx2. 

Eliminating  m  and  n  from  the  three  equations,  we  obtain 
ydx2  +  a?dy2  —  2xydxdy  -f  yx2d?y  =  0, 

which  is  the  general  differential  equation  of  lines  of  the 
second  order. 

98.  In  order  to  free  an  equation  of  its  constants,  it  will 
be  necessary  to  differentiate  it  as  many  times  as  there  are 
constants  to  be  eliminated.     For,  two  equations  are  neces- 
sary to  eliminate  a  single  constant,  three  to  eliminate  two 
constants,  four  to  eliminate  three  constants,  &c. :  hence, 
one  constant  may  be  eliminated  from*  the  given  equation 
and  the  first  differential  equation ;  two  from  the  given  equa- 
tion and  the  first  and  second  differential  equations,  &c. 

99.  The  differential  equation  which  is  obtained  after  the 
constants  are  eliminated,  belongs  to  a  species  or  order  of 
lines,  of  which  the  given  equation  represents  one  of  th 
species. 

Thus,  the  differential  equation  (Art.  94), 


DIFFERENTIAL    CALCULUS.  97 

belongs  to  an  order  or   species  of  lines   of  which  the 
equation 

y  =  ax  +  6, 

represents  a  single  one,  for  given  values  of  a  and  b. 
The  equation  of  a  parabola  is 


and  the  differential  equation  of  the  species  is 


yx  =    ,         or 

100.  The  differential  equation  of   a  species,   expresses 
the  law  by  which  the  variable  co-ordinates  change  their 
values  ;  and  this  equation  ought,  therefore,  to  be  indepen- 
dent of  the  constants  which  determine  the  magnitude,  and 
not  the  nature  of  the  curve. 

101.  The  terms  of  an  equation  may  be  freed  from  their 
exponents,  by  differentiating  the  equation  and  then  com- 
bining the  differential  and  given  equations. 

Suppose,  for  example, 

P"=Q, 

P  and  Q  being  any  functions  of  x  and  y. 
By  differentiating,  we  obtain 

nPn~ldP=dQ: 
by  multiplying  both  members  by  P,  we  have 


and  by  substituting  for  Pn  its  value, 


98  ELEMENTS    OF    THE 

The   same  result  -might  also   have   been  obtained  by 
taking  the  logarithms  of  both  members  of  the  equation 

Pn=Q. 

For,  we  have 


and  (Art.  57). 

dP      dQ 


hence,  nQdP  =  PdQ. 


Of  Vanishing  Fractions,  or  those  which  take  the 
form    ±. 

"^2.  It  has  been  shown  in  (Alg.  Art.  Ill),  that  -, 
though  a  symbol  of  an  undetermined  quantity,  may,  under 
particular  suppositions,  become  equal  to  0,  to  a  finite 
quantity,  or  to  infinity. 

This  symbol  arises  from  the  presence  of  a  common 
factor  in  the  numerator  and  denominator,  which,  becom- 
ing 0  for  a  particular  value  of  the  variable,  reduces  the 

fraction  to  the  form  -. 
0 

If  we  have,  for  example,  a  fraction  of  the  form 


Q(x  -  a)*' 

in  which  P  and   Q  are  functions  of  a?,  which  do  not  re- 
duce to  0,  for  x  —  a,  we  have 

.  P(x  -  a)m      0 
Q(x  -  a}n  ~~  0' 


DIFFERENTIAL    CALCULUS.  99 

The  value  of  this  fraction  will,  however,  be  0,  finite  or 
infinite,  according  as 

m  >  n,        m  =  n,         m  <  n, 

for  under  these  suppositions,  respectively,  it  takes  the  form 
P(x-a}m-n  J>  P 

~~3~  Q'          Q(*  -a)-"' 

Let  the  numerator  of  the  proposed  fraction  be  desig- 
nated by  X,  and  the  denominator  by  Xf,  and  let  us  sup- 
pose an  arbitrary  increment  h  to  be  given  to  x.  The 
numerator  and  denominator  will  then  become  a  function 
of  x  -f-  h,  and  we  shall  have  from  the  theorem  of  Taylor 

.  dX  h       #X  A2       cPX     h3          . 


dX'h  _  __ 

V  dx  T~h"3^~1.2  +  da*  1.2.3^ 

•  If  the  value  of  x  —  a,  reduces  to  0  the  differential 
coefficients  in  the  numerator  as  far  as  the  mth  order,  and 
those  of  the  denominator  as  far  as  the  wth  order,  the  value 
of  the  fraction  will  become, 

dmX      hm 


dxm   1.2.3.4...  . 


„ 


o 


dxn  1.2.3.4....7Z 

If  we  make  h  =  Q,  the  value  of  the  fraction  will  be- 
come 0,  finite,  or  infinite  according  as 

m  >  rc,         m  =  n,         m<n, 

and   hence,  if  the  value  x  =  a,  reduces   to   0  the  same 
number   of  differential  coefficients  in  the  numerator   and 


100  ELEMENTS    OF    THE 

denominator,  the  value  of  the  fraction  will  be  finite 
and  equal  to  the  ratio  of  the  first  differential  coefficients 
which  do  not  reduce  to  0. 

103.  Let  us  now  illustrate  this  theory  by  examples. 
1.  If  in  the  fraction 


we  make  x=  1,  we  have     — .  But 

dX  dXf 

—  =  -  nxn  \  —-  =  -  1 ; 

dx  dx 

in  which,  if  we  make  x  =  1 ,  we  have 

dX  dX' 

— — ——n,       and  — —  =  —  1, 

dx  dx 


hence> 


therefore,  the  value  of  the  fraction  when  x  =1,  is  +  n. 
2.  Find  the  value  of  the  fraction 

ax2  —  2  acx  + 


bx2- 


r,     when  a?  = 


=  2ax-2ac, 
dx  dx 

both   of  which  become   0,   when  x  —  c.     Differentiating 
again,  we  have 


_ 
x'~       '  x 

hence,  the  true  value  of  the  fraction  when  x  =  c  is    -7- 

o 


DIFFERENTIAL    CALCULUS.  101 

3.  Find  the  value  of  the  fraction 

x3  —  aa?  —  a2x  +  a3 

-£ 2 >     when  x  =  a. 

Ans.    0. 

4.  Find  the  value  of  the  fraction 


-,     when  x  =  a. 

Ans.     oo. 


5.  Find  the  value  of 


a*  —  b* 

-  ,     when  x  =  0. 
x 

Ans.     la  —  Ib. 

6.  What  is  the  value  of  the  fraction 

1  —  sin  07  +  cos  x 

-r—  —  ,     when  x  —  90°. 

sin  07  +  cos  a?  —  1 

Ans.     1. 

7.  What  is  the  value  of  the  fraction 

a  —  x  —  ala  +  alx 
-  7  —          _,     when  x  =  a. 
a-  V2ax~x2 

Ans.     —  1. 

8.  What  is  the  value  of  the  fraction 

x 

'—-,     when  x  =  1  . 


1  —  x  -f  Ix 

Ans.     —  2. 
9.  What  is  the  value  of  the  fraction 

an  —  xn 

7 7-,     when  x  —  a. 

la  —  Ix 

Ans.     ian. 


102  ELEMENTS    OF    THE 

104.  It  has  been  remarked  (Art.  47),  that  the  theorem 
of  Taylor  does  not  apply  to  the  case  in  which  a  particular 
value  attributed  to  oc,  renders  any  differential  coefficient 
of  the  iunction  infinite.  Such  functions  are  of  the  form 


(x-a  f  ' 

in  which  m  and  n  are  fractional. 

In  functions  of  this  form  we  substitute  for  a?,  a  +  h, 
which  gives  a  second  state  of  the  function.  We  then 
divide  the  numerator  and  denominator  by  h  raised  to  a 
power  denoted  by  the  smallest  exponent  of  h,  after  which 
we  make  h  =  0,  and  find  the  ratio  of  the  terms  of  the 
fraction. 

When  we  place  a  -f  h  for  a?,  we  have,  in  arranging 
according  to  the  ascending  powers  of  7^, 


F(a  +  h)  _    Ah"  +  Bhb  +  Che  +  &c., 

F(a  +  A)  ~  A'ha>  +  Bhb>  +  Clf  +  &c. 

Now  there  are  three  cases,  viz. :  when 

a  >  a! ,         a  —  afj         a  <  a1 '. 

In  the  first  case,  the  value  of  the  fraction  will  be  0 ;  in 
the  second,  a  finite  quantity ;  and  in  the  third  it  will  be 
infinite. 

105.  In  substituting  a-\-h  for  a?,  in  the  fraction 


(x-a)" 


DIFFERENTIAL    CALCULUS.  103 


,                         (2aft  +  ft2)2      ,0      .   M{- 
we  have  v § — *F  =  (2a  +  ft)2, 

ftT 

and  by  making  ft  =  0,  which  renders  x  =  a   the  value  of 
the  fraction  becomes 

3 


2.  Required  the  value  of  the  fraction 

* 

(a?-3ax+2a2)3 

l . — L —    when    x  = 

(a-'-o'F 

Substituting  a  -+-  ft  for  a?,  we  have 


ftT(3a2  +  3aft  +  ft2)2       (3a2  +  3aft+  ft2)2 

which  is  equal  to  0,  when  ft  =  0. 

106.  Remark.     The  last  method  of  finding  the  value  of 
a  vanishing  fraction,  may  frequently  be  employed  advan- 
tageously,   even  when   the  value   can  be   found   by  the 
theorem  of  Taylor. 

107.  There  are  several  forms  of  indetermination  under 
which  a  function  may  appear,  but  they  can  all  be  reduced 

to  the  form     — . 

0 
1st.    Suppose  the  numerator  and  denominator  of  the 

fraction 

JT 

to  become   infinite  by  the   supposition   of   sc  =  a.      The 
fraction  can  be  placed  under  the  form 


104  ELEMENTS    OF    THE 


which  reduces  to  — ,  when  X  and  Xf  are  infinite. 

2d.  We  may  have  the  product  of  two  factors,  one  of 
which  becomes  0  and  the  other  infinite,  when  a  particular 
value  is  given  to  the  variable. 

In  the  product  PQ,  let  us  suppose  that  x=a,  makes 
P  =  0  and  Q  =  oo  .  We  would  then  write  the  product 
under  the  form, 

P 

"Q" 

which  becomes  —  when  x  =  a. 

108.  Let  us  take,  as  an  example,  the  function 

(l-tf)tang  — *-o?; 

in  which  T  designates  180°. 

If  we  make  x—  1,  the  first  factor  becomes  0,  and  the 
second  infinite.  But 

1  1 

tang -•*•*?=:: ;       ^ 

pnf    ~prp 

COL     ^TTX 

hence,         (1  -  #)tang— *x  = =^- , 

cot  —  *x 

2 

the  value  of  which  is   —  when  x  =  1 . 


DIFFERENTIAL    CALCULUS.  105 


CHAPTER  V. 

Of  the  Maxima  and  Minima  of  a  Function  of  a 
Singh  Variable. 

109.  If  we  have 

*  =  /(*), 

the  value  of  the  function  u  may  be  changed  in  two  ways  : 
first,  by  increasing  the  variable  x ;  and  secondly,  by  dimin- 
ishing it. 

If  we  designate  by  u'  the  first  value  which  u  assumes 
when  x  is  increased,  and  by  u"  the  first  value  which  u 
assumes  when  x  is  diminished,  we  shall  have  three  con- 
secutive values  of  the  function 

!/,         u,         uff. 

Now,  when  u  is  greater  than  both  ur  and  urf,  u  is  said 
to  be  a  maximum :  and  when  u  is  less  than  both  u'  and 
u",  it  is  said  to  be  a  minimum. 

Hence,  the  maximum  value  of  a  variable  function  is 
greater  than  the  value  which  immediately  precedes,  or  the 
value  that  immediately  follows :  and  the  minimum  value 
of  a  variable  function  is  less  than  the  value  which  imme- 
diately precedes  or  the  value  that  immediately  folloivs. 

110.  Let  us  now  determine  the  analytical  conditions 
which  characterize  the  maximum  and  minimum  values  of 
a  variable  function. 


106  ELEMENTS    OF    THE 

If  in  the  function 

«  =  /(*), 

the  variable  x  be  first  increased  by  h,  and  then  diminished 
by  7i,  we  shall  have  (Art.  44), 

du  h        d2u    h2     .  d?u     h3         » 


T          __ 

h        d2u    h2       d?u    h3 


and  consequently, 

,  du  h       d*u    h2         (Pu      h3 

-w++ 


du  h       ffiu    h2         tfu      h3 

-  --- 


Now,  if  u  has  a  maximum  value,  it  will  be  greater 
than  u'  or  u"  ;  and  hence,  u'  —  u  and  u"  —  u  will  both 
be  negative.  If  u  is  a  minimum,  it  will  be  less  than  u' 
or  u"9  and  hence,  u'  —  u  and  u"  —  u  will  both  be  positive. 

Hence,  in  order  that  u  may  have  a  maximum  or  a 
minimum  value,  the  signs  of  the  two  developments  must 
be  both  minus  or  both  plus. 

But  since  the  terms  involving  the  first  power  of  h,  in 
the  two  developments,  liave  contrary  signs,  and  since  so 
small  a  value  may  be  assigned  to  h  as  to  make  the  first 
term  in  each  development  greater  than  the  sum  of  all  the 
other  terms  (Art.  44),  it  follows  that  u  can  have  neither 
a  maximum  nor  a  minimum,  unless 


dx 


DIFFERENTIAL    CALCULUS.  107 

and  the  roots  of  this  equation  will  give  all  the  values  of 
x  which  can  render  the  function  u  either  a  maximum  or 
a  minimum. 

Having  made  the  first  differential  coefficient  equal  to  0, 
the  signs  of  the  developments  will  depend  on  the  sign  of 
second  differential  coefficient. 

But  since  the  signs  of  the  first  members  of  the  equa- 
tions, and  consequently  of  the  developments,  are  both 
negative  when  u  is  a  maximum,  and  both  positive  when  u 
is  a  minimum,  it  follows  that  the  second  differential  co- 
efficient will  be  negative  when  the  function  is  a  maximum, 
and  positive  when  it  is  a  minimum.  Hence,  the  roots  of 
the  equation 

—  — 0 

being  substituted  in  the  second  differential  coefficient,  will 
render  it  negative  in  case  of  a  maximum,  and  positive  in 
case  of  a  minimum;  and  since  there  may  be  more  than  one 
value  of  x  which  will  satisfy  these  conditions,  it  follows 
that  there  may  be  more  than  one  maximum  or  one  minimum. 
But  if  the  roots  of  the  equation 

reduce  the  second  differential  coefficient  to  0,  the  signs 
of  the  developments  will  depend  on  the  signs  of  the 
terms  which  involve  the  third  differential  coefficient ;  and 
these  signs  being  different,  there  can  neither  be  a  maxi- 
mum nor  a  minimum,  unless  the  values  of  x  also  reduce 
the  third  differential  coefficient  to  0.  When  this  is  the 
case,  substitute  the  roots  of  the  equation 


]08  ELEMENTS    OF    THE 


in  the  fourth  differential  coefficient  ;  if  it  becomes  negative 
there  will  be  a  maximum,  if  positive  a  minimum.  If  the 
values  of  x  reduce  the  fourth  differential  coefficient  to  0, 
the  following  differential  coefficient  must  be  examined. 
Hence,  in  order  to  find  the  values  of  x  which  will  render 
the  proposed  function  a  maximum  or  a  minimum. 
1st.  Find  the  roots  of  the  equation 

^  =  0. 

dx 

2d.  Substitute  these  roots  in  the  succeeding  differential 
coefficients,  until  one  is  found  which  does  not  reduce  to  0. 
Then,  if  the  differential  coefficient  so  found  be  of  an  odd 
order,  the  values  of  x  will  not  render  the  function  either 
a  maximum  or  a  minimum.  But  if  it  be  of  an  even 
order,  and  negative,  the  function  will  be  a  maximum  ;  if 
positive,  a  minimum. 

111.  Remark.  Before  applying  the  preceding  rules  to 
examples,  it  may  be  well  to  remark,  that  if  a  variable 
function  is  multiplied  or  divided  by  a  constant  quantity, 
the  same  values  of  the  variable  which  render  the  function 
a  maximum  or  a  minimum,  will  also  make  the  product  or 
quotient  a  maximum  or  a  minimum,  and  hence  the  con- 
stant will  not  affect  the  conditions  of  maximum  or  mini- 
mum. 

2.  Any  value  of  the  variable  which  will  render  the* 
function  a  maximum  or  a  minimum,  will  also  render  any 
root  or  power  a  maximum  or  a  minimum  ;  and  hence,  if 
a  function  is  under  a  radical,  the  radical  may  be  omitted. 


DIFFERENTIAL    CALCULUS.  109 


EXAMPLES. 


1.  To  find  the  value  of  x  which  will  render  y  a  maxi- 
mum or  a  minimum  in  the  equation  of  the  circle 


dy  x 

' 


making     --  =  0,     gives  x  =  0. 
The  second  differential  coefficient  is 


da? 
and  since  making  x  =  0,  gives  y  —  R,  we  have 


dx*  ~        R 

which   being  negative,  the  value  of  x  =  0  renders  y   a 
maximum. 

2.  Find  the  values  of  x  which  will  render  y  a  maximum 
or  a  minimum  in  the  equation, 

y  =  a  —  bx  +  tf3, 
differentiating,  we  find 

^=-6  +  2*,      and      f|  =  2, 
ax  dor 

making,      —  b  +  2x  =  Q,     gives      d?  =  ~; 

2 

and  since  the  second  differential  coefficient  is  positive,  this 
value  of  x  will   render   y  a   minimum.     The  minimum 


110  ELEMENTS    OF    THE 

value  of  y  is  found  by  substituting  the  value  of  a?,  in  the 
primitive  equation.     It  is 


3.  Find  the  value  of  x  which  will  render  the  function 

u  =  a4  +  tfx  -  c  V, 

a  maximum  or  a  minimum, 

du      ,,      _  ,  b3 

fo  =  b    ~2CX'  X  =  -2<?' 

d?u  _  0 

and,  _—  =  —  2c2  : 

c^ar 

hence,  the  function  is   a  maximum,  and  the  maximum 
value  is 

•     1  u=a>+*     •     :  .  I 

4(T 

4.  Let  us  take  the  function 


we  find        -=-  =  9  «V  —  64,     and      a?  =  =b  — 
dx  3a 

The  second  differential  coefficient  is 

dzu  2 

—  =  18  A 

Substituting  the  plus  root  of  a?,  we  have 


DIFFERENTIAL    CALCULUS.  *  111 

which  gives  a  minimum,   and  substituting  the   negative 
root,  we  have 


which  gives  a  maximum. 

The  minimum  value  of  the  function  is, 


9a 
and  the  maximum  value 


112.  Remark.  It  frequently  happens  that  the  value 
of  the  first  differential  coefficient  may  be  decomposed  into 
two  factors,  X  and  X ',  each  containing  x,  and  one  of 
them,  X  for  example,  reducing  to  0  for  that  value  of  a?, 
which  renders  the  function  a  maximum  or  a  minimum. 
When  the  differential  coefficient  of  the  first  order  takes 
this  form,  the  general  method  of  finding  the  second  diffe- 
rential coefficient  may  be  much  simplified.  For,  if 

du 


we  shall  have 


dx  dx 


But  by  hypothesis  X  reduces  to  0  for  that   value   of  x 
which  renders  the  function  u  a  maximum  or  a  minimum  : 

d2u     XfdX 


112  »  ELEMENTS    OF    THE 

from  which  we  obtain  the  following  rule  for  finding  the 
second  differential  coefficient. 

Differentiate  that  factor  of  the  first  differential  coef- 
ficient which  reduces  to  0,  multiply  it  by  the  other  factor, 
and  divide  the  product  by  dx. 

5.  To  divide  a  quantity  into  two  such  parts  that  the  mih 
power  of  one  of  the  parts  multiplied  by  the  Tzth  power  of 
the  other  shall  be  a  maximum  or  a  minimum. 

Designate  the  given  quantity  by  a  and  one  of  the  parts 
by  a?,  then  will  a  —  x  represent  the  other  part.  Let  the 
product  of  their  powers  be  designated  by  u  ;  we  shall  then 
have 

u  =  xm(a  —  #)n, 

whence,       -r-  =  mxm~l  (a  —  x)n  —  nxm  (a  —  x)n~\ 

(J'OC 

=  (ma  —  mx  —  nx)xm~l  (a  —  a?)""1, 
and  by  placing  each  of  the  factors  equal  to  0,  we  have 
ma 


m  + 

The  second  differential  coefficient  corresponding  to  the 
first  of  these  values,  found  by  the  method  just  explained,  is 


and  substituting  for  x  its  value,  it  becomes 


(m  +  n) 


m-f-n  — 3 


hence,  this  value  of  x  renders  the  product  a  maximum. 
The  two  other  values  of  x  satisfy  the   equation   of  the 


DIFFERENTIAL    CALCULUS.  113 

problem,  but  do  not  satisfy  the  enunciation,  since  they  are 
not  parts  of  the  given  quantity   a. 

Remark.    If  m  and  n  are  each  equal  to  unity,  the  quan- 
tity will  be  divided  into  equal  parts. 

6.  To  determine  the  conditions  which  will  render  y  a 
maximum  or  a  minimum  in  the  equation 

y2  —  2mxy  +  a?  —  a2  =  Q. 
The  first  differential  coefficient  is 

dy  _  my  —  x  f 
dx     y  —  mx ' 

x 

hence,  my  —  x  =  0,       or      y  =  — . 

m 

Substituting  this  value  of  y  in  the  given  equation,  we 
find 

ma 

Vl-m2  ' 

and  the  value  of  y  corresponding  to  this  value  of  x  is 

a 


y  = 


To  determine  whether  y  is  a  maximum  or  a  minimum. 
let  us  pass  to  the  second  differential  coefficient.     We  have 


^y.    A 

hence,  d>y  =\fo~    ) 

dx2          y  —  mx 

8 


114  ELEMENTS    OF    THE 

and  since    ~  =  0,     we  have 
dx 


1 


dot?  y  —  mx* 

and  by  substituting  for  y  and  x  their  values,  we  have 


hence,  y  is  a  maximum. 

7.  To  find  the  maximum  rectangle  which  can  be  in- 
scribed in  a  given  triangle. 

Let  b  denote  the  base  of  the  triangle,  h  the  altitude, 
y  the  base  of  the  rectangle,  and  x  the  altitude.  Then, 

u=ixy  =  the  area  of  the  rectangle. 
But  b  :  h  :  :  y  :  h  —  x: 

bh  —  bx 
hence,  y  =  —  r  -  , 

and  consequently, 

bhx—  bxz       b  ,,         ,,v 
ti  =  -  -  -  =  —  (hx,-a?). 
h  h 

and  omitting  the  constant  factor, 

du      ^  h 

—  =  h-2x,       or       x  =  —  ; 
dx  2 

hence,  the  altitude  of  the  rectangle  is  equal  to  half  the 
altitude  of  the  triangle  :  and  since 


dx* 
the  area  is  a  maximum. 


DIFFERENTIAL    CALCULUS. 


115 


8.  What  is  the  altitude  of  a  cylinder  inscribed  in  a 
given  cone,  when  the  solidity  of  the  cylinder  is  a  maxi- 
mum ? 

Suppose  the  cylinder  to  be  inscribed, 
as  in  the  figure,  and  let 

AB  =  a,  BC  =  b,  AD  =  x,  ED  =  y  ;         EJ[~ 
then,   BD  =  a  —  x  =  altitude   of  the 
cylinder,    and     ny2(a  —  x)  =  solidity 

.;•.  (i) 

From    the    similar   triangles 
and  JlCB,  we  have 


bx 
— 


x  :  y  :  :  a  :  b  ;     whence     y  —  —  . 
Substituting  this  value  in  equation  (l),  and  we  have 


v  = 


a  -  x). 


f) 

Omitting  the  constant  factor  — £->   we  may  write 

u  =  x2(a  —  x) ; 

tor  the  conditions  which  will  make  u  a  maximum  will 
also  make  v  a  maximum  (Art.  111). 
By  differentiating,  we  have 

du  dzu 

-r-  =  2ax  —  3x2,    and    — j  =  2a  —  6x. 

dx  ay? 

Placing  2ax  —  3xz  =  0, 

we  have  x  =  0,     and     x  =  -a. 

.   *  3 

Hence  the  altitude  of  the  maximum  cylinder  is  one-third 
the  altitude  of  the  cone 


ELEMENTS    OF    THE 

9.  What  are  the  sides  of  the  maximum  rectangle  in- 
scribed in  a  given  circle  ? 

Ans.     Each  equal  to  R  V%. 

10.  A  cylindrical  vessel  is  to  contain  a  given  quantity 
of  water.     Required  the  relation  between  the  diameter  of 
the  base  and  the  altitude  in  order  that  the  interior  surface 
may  be  a  minimum. 

Ans.     Altitude  =  radius  of  base. 

11.  To  find  the  altitude  of  a  cone  inscribed  in  a  given 
sphere,  which  shall  render  the  convex  surface  of  the  cone 

a  maximum.  . 

Ans.     Altitude  =  —  R. 

12.  To  find  the  maximum  right-angled  triangle  which 
can  be  described  on  a  given  line. 

Ans.     When  the  two  §ides  are  equal. 

13.  What  is  the  length  of  the  axis  of  the  maximum 
parabola  that  can  be  cut  from  a  given  right  cone  ? 

Ans.     Three-fourths  the  side  of  the  cone. 

14.  To  find  the  least  triangle  which  can  be  formed  by 
the  radii  produced,  and  a  tangent  line  to  the  quadrant  of  a 
given  circle. 

Ans.  When  the  point  of  contact  is  at  the  middle  of  the 
arc. 

15.  What  is  the  altitude  of  the  maximum  cylinder  which 
can  be  inscribed  in  a  given  paraboloid  ? 

Ans.     Half  the  axis  of  the  paraboloid 


DIFFERENTIAL  CALCULUS. 


116* 


CHAPTER   VI. 

Application   of  the   Differential    Calculus  to   the 
Theory  of  Curves. 

113.  Every  relation  between  a  function  y  and  a  van- 
able  x,  expressed  by  the  equation 

y  =/(*), 

will  subsist  between  the  ordinate  and  abscissa  of  a  curve 

Let  Ji  be   the  origin  of  the  rectangular  axes ;  then 

if  in  the  equation  , 

\C 
we  make  x  =  0,  we  have 

y  —  a  constant : 
lay  off  JIB  equal  to  this  con- 
slant.    Then  attribute  values  to 
a?,  from  0  to  any  limit,  as  well  T      A  H 

negative  as  positive,  and  find  from  the  equation 

the  corresponding  values  of  y.  Conceive  the  values  of  x 
to  be  laid  off  on  the  axis  of  abscissas,  and  the  values  of 
y  on  the  corresponding  ordinates.  The  curve  described 
through  the  extremities  of  the  ordinates  will  have  for  its 
equation  y=f(x\  (0 

Let  x  represent  any  abscissa,  J1H  for  example,  and 
y  the  corresponding  ordinate  HP. 

If  now  we  give  to  x  any  arbitrary  increment  h,  and 
make  HF  —  h,  the  value  of  y  will  become  equal  to  FC, 
which  we  will  designate  by  y'.  Through  P  draw  the 
secant  CPI  and  the  tangent  TP. 


116s 


ELEMENTS    OF    THE 


Now,  y'-y=CF-PH=  CD, 

CD 

T2' =  Pi)  ~  tansent  of  tlie  ansle  CPD- 

CD      PH 


y>-y      CD 
and ~ 


But,  by  similar  triangles 

Now,  the  limiting  ratio  of  the  increment  of  the  variable 
to  that  of  the  function,  is  that  ratio  which  is  independent 
of  the  value  of  /z,  and  is  obtained  by  making  h  equal  to  0 
in  the  expression  for  the  ratio  of  the  increments  (Art.  15.) 

It  is  evident  that  as  h  diminishes,  the  point '  C  will  ap- 
proach the  point  P,  the  point  /  will  approach  T,  and  the 
secant  1C  will  approach  the  tangent  TP ;  and  when  h 
becomes  equal  to  0,  the  secant  1C  will  coincide  with  the 
tangent  TP.  For  every  position  of  C  we  shall,  have 

C1  T)       PTT 
~pj)~~j^f=  tangent  CPD  =  tangent  CIH ;  .and  when 

C  coincides  with  P,  ~W=~==  tangent  PTH—    -; 


that  is,  the  limiting  ratio,  or  first  differential  coefficient, 
is  equal  to  the  tangent  of  the  angle  which  the  tangent 
line  makes  with  the  axis  of  abscissas. 

Of  Tangents  and  Normals. 

114.  Having  found  the  value  of 

dy 

dx 

we  will  now  proceed  to  find  the 
value  of  the  subtangent,  tangent, 
subnormal,  and  normal.  T  A  R  N 


DIFFERENTIAL    CALCULUS.  117 

We  have        (Trig.  Th.  II), 

1   :   TR  ::  tangT  :  RP; 
that  is,  1  :   TR  ::      -j-      :  y 

dx 
hence,  TR  —y—  —  sub-tangent. 

115.  The  tangent    TP  is  equal  to  the   square  root  of 
the  sum  of  the  squares  of  TR  and  RP ;  hence, 

TP  =  y  V  1  +  j-a  =  tangent. 

116.  From  the  similar  triangles   TPR,  RPN,   we  have 

TR  :  PR  ::  PR  :  RN, 

(i  T 

hence,  y  —    :     y     :  :     y     :  RN, 

consequently,  RN  =  y-¥=  sub-normal. 

1 17.  The  normal  PN  is  equal  to  the  square  root  of  the 
sum  of  the  squares  of  PR  and  RN ;  hence, 


=  y\/\  +-TZ  = 


normal. 


118.  Let  it  be  now  required  to  apply  these  formulas  to 
lines  of  the  se.cond  order,  of  which  the  general  equation 
(An.  Geom.  Bk.  VI,  Prop.  XII,  Sch.  3),  is, 

y2  —  mx  -f  nx2. 
Differentiating,  we  have 

dy  _m-\-  2nx  _     m 


118 


ELEMENTS    OF    THE 


substituting  this  value,  we  find 

,    .  ™D         dx 

sub-tangent  TR  =  y-~  = 
y  dy 


. 

2nx 


-  =  A/  mx  +  fta?2  +  4 


mx 


iin: 


!T 


i  r>  AT        dy      m  -\-2nx 
sub-normal  RN  —  y-Z  =  -  , 
y  dx  2 


PN 


=  y\J  \  +  %£  =y 


nx2  -f-  —  (m 


By  attributing  proper  values  to  m  and  ??,  the  above 
formulas  will  become  applicable  to  each  of  the  conic 
sections.  In  the  case  of  the  parabola,  n  =  0,  and  we  have 


W=f- 


PN 


=  \/  mx  +  —  m2. 


119.  It  is  often  necessary  to  represent  the  tangent  and 
normal  lines  by  their  equations.  To  determine  these,  in 
a  general  manner,  it  will  be  necessary  first  to  consider  the 
analytical  conditions  which  render  any  two  curves  tangent 
to  each  other. 

Let  the  two  curves,  PDC, 
PEC,  intersect  each  other  at 
P  and  C. 

Designate  the  co-ordinates  of 
the  first  curve  by  x  and  y,  and 
the  co-ordinates  of  the  second  by 
a/,  y'.  Then,  for  the  common 
point  Py  we  shall  have 

x  ~  a/,         y  =  yf. 


DIFFERENTIAL    CALCULUS.  ]  19 

ff  we  represent  BG,  the  increment  of  the  abscissa,  by  h, 
we  shall  have,  from  the  theorem  of  Taylor  (Art.  44), 

<*-»-«*•      + 


hence,  by  placing  the  two  members  equal  to  each  other, 
and,  dividing  by  h,  we  have 

cfy      J2y   h  dy'        d?y'    h 

+  +  &c"  =  - 


If  we  now  pass  to  the  limit,  by  making  ^  =  0,  we  shall 
have 

dy        dy' 
fa  =  ~djT; 

in  which  case  the  point  C  will  become  consecutive  with  P, 
and  the  curve  PEC  tangent  to  the  curve  PDC.  Hence, 
two  lines  will  be  tangent  to  each  other,  when  they  have 
a  common  point,  and  the  first  differential  coefficient  of 
the  one  equal  to  the  first  differential  coefficient  of  the 
other,  for  this  point. 

120.  The  equation  of  a  straight  line  is  of  the  form 
y  =  ax  +  b, 

dy 
hence,  ^  =  a. 

But  the  equation  of  a  straight  line  passing  through  a 
given  point,  of  which  the  co-ordinates  are  a?/x,  y",  is  (An. 
Geom.  Bk.  II,  Prop.  IV), 

y-y"=a(x-x"). 


120  ELEMENTS    OF   THE 

. 

But  if  the  point  whose  co-ordinates  are  a?",  y'',  is  required 
to  be  on  a  given  curve,  these  co-ordinates  must  satisfy 
the  equation  of  that  curve.  If  the  straight  line  is  required 
to  be  tangent  to  the  curve  at  this  particular  point,  the 

first  differential  coefficient    --^,  found  from  the    equation 

doc 

du" 
of  the  curve,  must  take  the  particular  value    -g—/  ;  that  is, 

.  G/Ou 

we  must  have 

dy_dy^ 

dx  ~  da? 

and  the  equation  of  the  line  tangent  at  the  point  whose 
co-ordinates  are  a/',  y",  will  be 


121.  Let  it  be  required,  for  example,  to  make  the  line 
tangent  to  the  circumference  of  a   circle   at  a  point  of 
which  the  co-ordinates  are  x",  y". 
The  equation  of  the  circle  is  x2  +  y2  =  R2  ; 

dy          x 
and,  by  differentiating,  we  have    •—  =  --  . 

But  if  the  straight  line  is  to  be  tangent  to  the  circle,  at 
the  point  whose  co-ordinates  are  #",  y",  we  must  have 


dx"  ~  dx~~       y  y// 

and  by  substituting  this  value  in  the  equation  of  the  line, 
and  recollecting  that  sc"2  +  y"2  =  R2,  we  have 

yy"  +  xx"  =  R2, 
which  is  the  equation  of  a  tangent  line  to  a  circle. 

122.  A  normal  line  is  perpendicular  to  the  tangent  at 


DIFFERENTIAL    CALCULUS.  121 

the  point  of  contact,  and  since  the  equation  of  the  tangent 
is  of  the  form 


the  equation  of  the  normal,  at  the  point  whose  co-ordin- 
ates are  oc",  y",  will  be  of  the  form  (An.  Geom.  Bk.  II., 
Prop.  VII.,  Sch.  2), 


y 

If  we  take  the   equation  of  any  curve,   and    find   the 

value  of    —  -    for  the  particular  point  whose  co-ordinates 
ay 

are  a/',  y",  and  then  substitute  thac  value  in  the  above 
equation,  we  shall  have  the  equation  of  the  normal  pas- 
sing through  this  point. 

The  equation  of  th*e  normal  in  the  circle  will  take  the  form 


123.  To  find  the  equation  of  a  tangent  line  to  an  ellipse 
at  a  point  of  which  the  co-ordinates  are  #",  y",  we  have 


By  differentiating,  we  have 


hence,  we  have 

l-rf' 

*-*'),  or 


and  for  the  normal      y  —  y"  =  iSuyX*  ~~  x" 


122 


ELEMENTS    OF    THE 


124.  To  find  the  equation  of  a  tangent  to  lines  of  the 
second  order,  of  which  the  equation  for  a  particular  point 
(An.  Geom.  Bk.  VI,  Prop.  XII,  Sch.  3)  is 


By  differentiating,  we  have 


d  2y'f 

hence,  the  equation  of  the  tangent  to  a  line  of  the  second 
order  is 


,,      m  +  2na/'  ,          /A 

y-y"=     8y//    (*-«"). 


and  the  equation  of  the  normal 

y-y"=-     2y 


Of  Asymptotes  of  Curves. 

125.  An  asymptote  of  a  curve  is  a  line  which  continually 
approaches  the  curve,  and  becomes  tangent  to  it  at  an 
infinite  distance  from  the  origin  of  co-ordinates. 

Let  AX  and  AY  be 
the  co-ordinate  axes,  and 


the  equation  of  any  tan 
gent  line,  as  TP. 


R 


T 


DIFFERENTIAL    CALCULUS.  123 

If  in  the  equation  of  the  tangent,  we  make  in  succes- 
sion y  —  0,  x—  0,  we  shall  find 


If  the  curve  CPB  has  an  asymptote  RE,  it  is  plain 
that  the  tangent  PT  will  approach  the  asymptote  RE, 
when  the  point  of  contact  P,  is  moved  along  the  curve 
from  the  origin  of  co-ordinates,  and  T  and  D  will  also 
approach  the  points  R  and  Y,  and  will  coincide  with 
them  when  the  co-ordinates  of  the  point  of  tangency  are 
infinite. 

Jn  order,  therefore,  to  determine  if  a  curve  have  asymp- 
totes, we  substitute  in  the  values  of  AT  and  AD,  the  co- 
ordinates of  the  point  which  is  at  an  infinite  distance  from 
the  origin  of  co-ordinates.  If  either  of  the  distances  AT, 
AD,  become  finite,  the  curve  will  have  an  asymptote. 

If  both  the  values  are  finite,  the  asymptote  will  be  in- 
clined to  both  the  co-ordinate  axes  :  if  one  of  the  distances 
becomes  finite  and  the  other  infinite,  the  asymptote  will 
be  parallel  to  one  of  the  co-ordinate  axes  ;  and  if  they  both 
become  0,  the  asymptote  will  pass  through  the  origin  of 
co-ordinates.  In  the  last  case,  we  shall  know  but  one 
point  of  the  asymptote,  but  its  direction  may  be  deter- 

minedly finding  the  value  of   -^,    under  the  supposition 
that  the  co-ordinates  are  infinite. 

126.  Let  us  now  examine  the  equation 

\f  =  mx  +  no?, 


124  ELEMENTS    OF    THE 

of  lines  of  the  second  order,  and  see  if  these  lines  have 
asymptotes.     We  find 


A  n  _ 

which  may  be  put  under  the  forms 


and  making  x  —  oo  ,  we  have 


AR=-—          and 


If  now  we  make  n  =  0,  the  curve  becomes  a  parabola, 
and  both  the  limits,  AR,  AE,  become  infinite  :  hence, 
the  parabola  has  no  rectilinear  asymptote. 

If  we  make  n  negative,  the  curve  becomes  an  ellipse, 
and  AE  becomes  imaginary:  hence,  the  ellipse  has  no 
asymptote. 

But  if  we  make  n  positive,  the  equation  becomes  that 

of  the  hyperbola,  and  both  the  values,  ARf  AE,  become 

B2 

finite.     If  we  substitute  for  n  its  value  —^  we  ?hall  have 

A. 

AR=  -A,        and        AE  =  ±  B. 


DIFFERENTIAL    CALCULUS. 


125 


Differentials  of  tJie  Arcs  and  Areas  of  Segments 
of  Curves. 


7.  It  is  plain,  that  the  chord  and  arc  of  a  curve  will 
approach  each  other  continually  as  the  arc  is  diminished, 
and  hence,  we  might  conclude  that  the  limit  of  their  ratio 
is  unity.  But  as  several  propositions  depend  on  this  rela- 
tion between  the  arc  and  chord,  we  shall  demonstrate  it 
rigorously. 

128.  If  we  suppose  the  ordi- 
nate  PR  of  the  curve,  POM  to 
be  a  function  of  the  abscissa,  we 
shall  have  (Art.  19), 

:  h.          and 


N 


Ji 


in  which 


=  -f- 
dx 


Hence,       PM=  Vh2+(P+P/h)2tf=hVl+(P+P'h?. 
We  also  have  NQ  =  Ph ; 


hence, 


PN=Vh2  +  P2h2  = 

NM  =  NQ-MQ=-  P'tf- 
hence,  we  have 

PN  +  MN     hVl+F2-P'h2 


PM 


hVl+(P  +  P'h)2      Vl  +  (P  + 


J26  ELEMENTS    OF    THE 

of  which  tho  limit,  by  making  h  —  0,  is 

IT 


But  the  arc  POM  can  never  be  less  than  the  chord  PM, 
nor  greater  than  the  broken  line  PNM  which  contains  it  ; 
hence,  the  limit  of  the  ratio 

POM, 

~¥M~ 

and  consequently,  the  differential  of  the  arc  is  equal  to 
the  differential  of  the  chord.  If  we  designate  the  arc  by  z, 
PM  will  be  represented  by  z/  —  z,  and  we  shall  have 


"h  PM       PQ~     PM       7, 

and,  by  passing  to  the  limiting  ratio, 


or  z 

that  is,  the  differential  of  the  arc  of  a  curve,  at  any  point, 
is  equal  to  the  square  root  of  the  sum  of  the  squares  of 
the  differentials  of  the  co-ordinates. 

}  29.  To  determine  the  differential  of  the  arc  of  a  circle 
of  which  the  equation  is 


xdx 
we  have   xdx  +  ydy  =  0,      or      ay  =  -- 


whence,   dz  = 


DIFFERENTIAL    CALCULUS.  127 


Rdx  _  Rdx 

i_   — 1— 


7 


y 

the  same  as  determined  in  (Art  71).  The  plus  sign  is  to 
be  used  when  the  abscissa  x  and  the  arc  are  increasing 
functions  of  each  other,  and  the  minus  sign  when  they 
are  decreasing  functions  (Art.  31). 

130.  Let  BCD  be  any  segment 
of  a  curve,  and  let  it  be  required 
to  find  the  differential  of  its  area. 

The     two     rectangles       DCFE,        B/ 
DOME,    having    the    same    base 
DE,   are  to   each  other  as  DC  to 
EM;  and  hence,  the  limit  of  their 
ratio  is   equal  to  the  limit  of  the  ratio  of  DC  to  EM, 
which  is  equal  to  unity. 

But  the  curvelinear  area  DCME  is  less  than  the  rect- 
angle DOME,  and  greater  than  the  rectangle  DCFE  : 
hence,  the  limit  of  its  ratio  to  either  of  them  will  be 
unity.  But, 

DCME   DCME   DEFC       DCME 
— •<* Tji  v  

DE  DE          DEFC  "  DEFC' 

or  by  representing  the  area  of  the  segment  by  s  and  the 
ordinate  DC  by  y,  and  passing  to  the  limit,  we  have 

ds  7/7 

—  =r  y,         or         ds  =  yax ; 

hence,  the  differential  of  the  area  of  a  segment  of  any 
curve,  is  equal  to  the  ordinate  into  the  differential  of  the 
abscissa. 


128  ELEMENTS    OF   THE 

131.  To  find  the  differential  of  the  area  of  a  circular 
segment,  we  have 

a?  -{-  y1  =  jR2,         and  y  =  VR2  —  a?  ; 

hence,  ds  =  dx  VR2  —  a?. 

The  differential  of  the  segment  of  an  ellipse,  is 


and  of  the  segment  of  a  parabola 

ds  —  dx  V2px. 

Signification  of  the  Differential  Coefficients. 

132.  It  has  already  been  shown  that,  if  the  ordinate  of 
a  curve  be  regarded  as  a  function  of  the  abscissa,  the  first 
differential  coefficient  will  be  equal  to  the  tangent  of  the 
angle  which  the  tangent  line  forms  with  the  axis  of  abscis- 
sas (Art.  1 13).  We  now  propose  to  show  the  signification 
of  the  second  differential  coefficient,  the  ordinate  being  re- 
garded as  a  function  of  the  abscissa. 

Let  AP  be  the  abscissa 
and  PM  the  ordinate  of  a 
curve.  From  P  lay  off 
on  the  axis  of  abscissas 
PP'  =  h,  and  PP"  =  2h. 
Draw  the  ordinates  PM, 
P'M.P"M";  also  the  lines 
MMN,  MM";  and  lastly, 
MQ,  M'Q',  parallel  to  the 


P' 


P" 


DIFFERENTIAL    CALCULUS. 


129 


axis  of  abscissas.     Then  will  M'Q  =  NQf,  and  we  shall 
have 

PM=y, 


-  M'Q 


1.2 


&c. 


Now,  since  the  sign  of  the  first  member  of  the  equation 
is  essentially  positive,  the  sign  of  the  second  member  will 
also  be  positive  (Alg.  Art.  85).  But  by  diminishing  h,  the 
sign  of  the  second  member  will  depend  on  that  of  the 
second  differential  coefficient  (Art.  44) :  hence,  the  second 
differential  coefficient  is  positive. 


If  the  curve  is  below 
the  axis  of  abscissas, 
the  ordinates  will  be 
negative,  and  it  is  easily 
seen  that  we  shall  then 
have 


130 


ELEMENTS    OF    THE 


Now,  since  the  first  metnber  is  negative,  the  second 
member  will  be  negative  :  hence  we  conclude  that,  if  a 
curve  is  convex  towards  the  axis  of  abscissas,  the  ordi- 
nate  and  second  differential  coefficient  will  have  like  signs. 

N 


133.  Let  us  now  con- 
sider the  curve  CMM'M" , 
which  is  concave  towards 
the  axis  of  abscissas.  We 
shall  have, 

p*n=v, 


M 


M 


p, 


-  P'M  = 


- 
dx 


M'fQ!-  MfQ  =  -  NM'= 


dor 


dor   1  .  2 


&c. 


But  since  the  first  member  of  the  equation  is  negative, 
the  essential  sign  of  the  second  member  will  also  be 
negative  :  hence,  the  second  differential  coefficient  will 
be  negative. 


DIFFERENTIAL    CALCULUS. 


131 


P' 


If  the  curve  is  below  the 
axis  of  abscissas,  the  ordi- 
nate  will  be  negative,  and  it 
is  easily  seen  that  we  should 
then  have 


M"Q!-  MfQ  =  +  NM"  = 


oar 


&c.  ; 


hence  we  conclude  that,  if  a  curve  is  concave  towards  the 
axis  of  abscissas,  the  ordinate  and  second  differential 
coefficient  will  have  contrary  signs. 

The  ordinate  will  be  considered  as  positive,  unless  the 
contrary  is  mentioned. 

134.  Remark  1.  The  co-ordinates  x  and  y,  determine 
a  single  point  in  a  curve,  as  M.     The  differential  of  y  is 
derived  from  the  ordinate  PM,  and  is  what  QM'  becomes 
when  the  ordinates  P'Jtf'  and  PM  become  consecutive. 

The  second  differential  of  y  is  derived  from  JWPQ,  in 
the  same  way  that  dy  is  derived  from  the  primitive  func- 
tion y.  ^It  is,  indeed,  what  JVf'Q'  becomes,  when  M"Q' 
becomes  consecutive  with  JV/'Q.  The  abscissa  x  being 
supposed  to  increase  uniformly,  the  difference  between 
PP'  and  P'P"  is  0  :  and  therefore  the  second  differential 
of  x  is  0.  The  co-ordinates  x  and  y,  and  the  first  and 
second  differentials  determine  three  points,  M,  M',  M", 
consecutive  with  each  other. 

135.  Remark   2.     When   the  curve  is  convex  towards 


132  ELEMENTS    OF    THE 

the  fcxis  of  abscissa,  the  first  differential  coefficient,  which 
represents  the  tangent  of  the  angle  formed  by  the  tangent 
line  with  the  axis  of  abscissas,  is  an  increasing  function 
of  the  abscissa  :  hence,  its  differential  coefficient,  that  is, 
the  second  differential  coefficient  of  the  function,  ought 
to  be  positive  (Art.  31). 

When  the  curve  is  concave,  the  first  differential  coeffi- 
cient is  a  decreasing  function  of  the  abscissa  ;  hence,  the 
second  differential  coefficient  should  be  negative  (Art.  31  ). 

Examination  of  the  Singular  Points  of  Curves. 

136.  A  singular  point  of  a  curve  is  one  which  is  dis 
tinguished  by  some  particular   property   not  enjoyed  by 
the  points  of  the  curve  in  general. 

Let  us,  as  a  first  example,  find  the  points  of  a  curve, 
through  which  the  tangent  lines  will  be  parallel  or  per- 
pendicular to  the  axis  of  abscissa* 

137.  Since  the  first  differential  coefficient  expresses  the 
value  of  the  tangent  of  the  angle  which  the  tangent  line 
forms  with  the  axis  of  abscissas,  and  since  the  tangent  is 
0,  when  the  angle  is  0,  and  infinite  when  the  angle  is  90°, 
it  follows  that  the  roots  of  the  equation 


will  give  the  abscissas  of  all  the  points  at  which  the  tan- 
gent is  parallel  to  the  axis  of  abscissas,  and  the  roots  of 
the  equation 

dy  dx 

-2-  =  oo  ,         or         —  =  0, 

dx  dy 


DIFFERENTIAL    CALCULUS.  133 

will  give  the  abscissas  of  all  the  points  at  which  the  tan- 
gent is  perpendicular  to  the  axis  of  abscissas. 

138.  If  a  curve  from  being  convex  towards  the  axis  of 
abscissas  becomes  concave,  or  from  being  concave  becomes 
convex,  the  point  at  which  the  change  of  curvature  takes 
place  is  called  a  point  of  inflexion. 

Since  the  ordinate  and  differential  coefficient  of  the 
second  order  have  the  same  sign  when  the  curve  is  convex 
towards  the  axis  of  abscissas,  and  contrary  signs  when  it 
is  concave,  it  follows  that  at  the  point  of  inflexion,  the 
second  differential  coefficient  will  change  its  sign.  There 
fore  between  the  positive  and  negative  values  there  will  be 
one  value  of  x  which  will  reduce  the  second  differential 
coefficient  to  0  or  infinity  (Alg.  Art.  310)  :  hence  the  roots 
of  the  equations 


will  give  the  abscissas  of  the  points  of  inflexion. 

139.  Let  us  now  apply  these  principles  in  discussing 
the  equation  of  the  circle 


We  have,  by  differentiating, 

dy  _       x 

dx~  ~y 

and  placing 

/yi 

--  =  0,       we  have       x  =  0. 

y 

Substituting  this  value  in  the  equation  of  the  curve,  we 
have 

y  =  ±  R  ; 


134  ELEMENTS    OF    THE 

hence,  the  tangent  is  parallel  to  the  axis  of  abscissas  at 
the  two  points  where  the  axis  of  ordinates  intersects  the 
circumference. 
If  we  make 

dy  x  y 

JL  =  --  =  QD  or         -  -i-  =  0, 

dx  y  x 

we  have  y  —  0  ;  substituting  this  value  in  the  equation, 
we  find 

x=  ±R, 

and  hence,  the  tangent  is  perpendicular  to  the  axis  of 
abscissas  at  the  points  where  the  axis  intersects  the  cir- 
cumference. 

The  second  differential  coefficient  is  equal  to 


which  will  be  negative  when  y  is  positive,  and  posrtive 
when  y  is  negative.  Hence,  the  circumference  of  the 
circle  is  concave  towards  the  axis  of  abscissas. 

If  we  apply  a  similar  analysis  to  the  equation  of  the 
ellipse,  we  shall  find  the  tangents  parallel  to  the  axis  of 
abscissas  at  the  extremities  of  one  axis,  and  perpendicular 
to  it  at  the  extremities  of  the  other,  and  the  curve  concave 
towards  its  axes. 

140.  Let  us  now  discuss  a  class  of  curves,  which  may 
be  represented  by  the  equation 

y  =  b±c(x  —  a)m, 

in  which  we  suppose  c  To  be  positive  or  negative,  and 
different  values  to  be  attributed  to  the  exponent  m. 


DIFFERENTIAL    CALCULUS. 


135 


1st.   When  c  is  positive,  and  m  entire  and  even. 

By  differentiating,  we  have 

dy  =  r_, 

dx      m>  ' 


- 
dor 


If  we  place  the  value    -~  =  0,  we  find  x  =  a,  and  sub- 
dx 

stituting  this  value  in  the  equation  of  the  curve,  we  find 


hence,  x  —  a,  y  —  6,  are  the  co-ordinates  of  the  point 
at  which  the  tangent  line  is  parallel  to  the  axis  of 
abscissas. 

Since  m  is  even,  m  —  2  will 
also  be  even,  and  hence  the  second  V 
differential  coefficient  will  be  posi- 
tive for  all  values  of  x.  The  curve 
will  therefore  be  convex  towards 
the  axis  of  X,  and  there  will  be 
no  point  of  inflexion. 

The  value  of  x  —  a  renders  the  ordinate  y  a  minimum, 
since  after  m  differentiations  a  differential  coefficient  of  an 
even  order  becomes  constant  and  positive  (Art.  110). 

The  curve  does  not  intersect  the  axis  of  X,  but  cuts  the 
axis  of  Y  at  a  distance  from  the  origin  expressed  by 

y  =  b  -f-  cam. 


136 


ELEMENTS    OF    THE 


141.    2d.  When  c  is  negative,  and  m  entire  and  even. 
We  shall  have,  by  differentiating,    y  =  b  —  e(x—a)m 


. 

dx 


and 


The  discussion  is  the  same  as 
before,  excepting  that  the  second 
differential  coefficient  being  nega- 
tive for  all  values  of  x,  the  curve 
is  concave  towards  the  axis  of 
abscissas,  and  the  value  of  x  =  a, 
renders  the  ordinate  y  a  maxi- 
mum (Art.  110). 

142.     3d.   When  c  is  plus  or  minus,  and  m  entire  and 
uneven. 

We  shall  have,  by  differentiating, 

dy  ,          \nT_i 

--._  ±mc(x-a)      , 


and 


The  first  differential  coefficient  will  be  0,  when  x  —  a , 
hence,  the  tangent  will  be  parallel  to  the  axis  of  abscissas, 
at  the  point  of  which. the  co-ordinates  are  x  =  a,  y  =  b 


DIFFERENTIAL  CALCULUS.  137 

Since  tfie  exponent  m  —  2  is 
uneven,  the  factor  (x  —  a)m~  2  will 
be  negative  when  #  <  a,  and 
positive  when  #  >  a;  hence,  this 
factor  changes  its  sign  at  the 
point  of  the  curve  of  which  the  _ 
abscissa  is  x  =  a. 

If  c  is  positive,  the  second  differential  coefficient  will  be 
negative  for  #<a,  and  positive  for  x>a:  hence  there  will 
be  an  inflexion  when  x  =  a.  If  c  were  negative,  the  curve 
would  be  first  convex  and  then  concave  towards  the  axis 
of  abscissas,  but  there  would  still  be  an  inflexion  at  the 
point  x  —  a.  At  this  point  the  tangent  line  separates  the 
two  branches  of  the  curve. 

There  will,  in  this  case,  be  neither  a  maximum  nor  a 
minimum,  since  after  m  differentiations  a  differential  coef- 
ficient of  an  odd  order,  will  become  equal  to  a  constant 
quantity  (Art.  110). 

143.     4th.   When  c  is  positive  or  negative,  and  m  a 

o 

fraction  having  an  even  numerator,  as  m  =  —  . 

o 

By  differentiating,  and  supposing  c  positive,  we  have 

dy       2     ,          ,4-i  2c 

-J  =  irc(*-a)'       =  -  -  -p 

3(x  —  a)3 
d*  2c 


If  we  make  x  =  a,  the  first  differential  coefficient  will 
become  infinite  ;  and  the  tangent  will  be  perpendicular  to 


138 


ELEMENTS    OF    THE 


the  axis  of  abscissas,  at  the  point  of  which  the  co-ordinates 
are  x  =  a,  y  =  b. 

In  regard  to  the  second  differen- 
tial coefficient,  it  will  become  infi- 
nite for  x  =  a,  and  negative  for 
every  other  value  of  x,  since  the 
factor  (x  —  a)  of  the  denominator 
is  raised  to  a  power  denoted  by  an 
even  exponent.  Hence,  the  curve 
will  be  concave  towards  the  axis  of 
abscissas. 

If  we  take  the  equation  of  the  curve 

£ 

y  =  b  +  c(x  —  a)3, 

and  make   x  =  a  +  h,    and  x  =  a  —  h,  we  shall  have,  m 
either  case, 


and  hence,  y  will  be  less  for  x  =  a,  than  for  any  other 
value  of  x,  either  greater  or  less  than  a.  Hence,  the 
value  x  =  a,  renders  y  a  minimum. 

If  c  were  negative,  the  equation  would  be  of  the  form 

j  \          / .  i 

and  we  should  have,  by  differentiating, 

dy  _  2c 

dx~~  T 


and 


2c 


da? 


DIFFERENTIAL  CALCULUS. 


139 


The  first  and  second  differen- 
tial coefficients  will  be  infinite  for 
x  =  a,  and  the  second  differential 
coefficient  will  be  positive  for  all 
values  of  x  greater  or  less  than  a; 
and  hence,  the  curve  will  be  con- 
vex towards  the  axis  of  abscissas. 

If,  in  the  equation  of  the  curve 
y  =  b  —  c(x — a] 


we  make  x 
either  case, 


a  +  h,  and  x  —  a  —  A,  we  shall  have,  in 

y  =  b  —  c Jfi  ; 

and  hence,  y  will  be  greater  for  x—  a,  than  for  any  other 
value  of  x  either  greater  or  less  than  a.  Hence,  the 
value  x  =  «,  renders  y  a  maximum. 

144.  Remark.  The  conditions  of  a  maximum  or  a 
minimum  deduced  in  Art.  110,  were  established  by  means 
of  the  theorem  of  Taylor.  Now,  the  case  in  which  the 
function  changes  its  form  by  a  particular  value  attri- 
buted to  x,  was  excluded  in  the  demonstration  of  that 
theorem  (Art.  45).  Hence,  the  conditions  of  minimum 
and  maximum  deduced  in  the  two  last  cases,  ought 
not  to  have  appeared  among  the  general  conditions  of 
Art.  110. 

We  therefore  see  that  there  are  two  species  of  maxima 
and  minima,  the  one  ckaracterized  by 


=  0,      the  .other  by 
dx  J       dx 


140  ELEMENTS  OP  THE 

In  the  first,  we  determine  whether  the  function  is  a 
maximum  or  a  minimum  by  examining  the  subsequent 
differential  coefficient  ;  and  in  the  second,  by  examining 
the  value  of  the  function  before  and  after  that  value  of  x 
which  renders  the  first  differential  coefficient  infinite. 

The  branches  DE,  ME,  which  are  both  represented  by 
the  equation. 

y  =  b  =fc  c(x  —  a)7, 

are  not  considered  as  parts  of  a  continuous  curve.  For, 
the  general  relations  between  y  and  x  which  determine 
each  of  the  parts  DE,  ME,  is  entirely  broken  at  the 
point  My  where  x~a.  The  two  parts  are  therefore 
regarded  as  separate  branches  which  unite  at  M.  The 
point  of  union  is  called  a  cusp,  or  a  cusp  point. 

145.     5th.    When  c  is  positive  or  negative  and  m  a 

3 

fraction  having  an  even  denominator,  as  m  =-j. 

Under  this  supposition  the  equation  of  the  curve  will 
become 


and  by  differentiating,  we  have 

d  3c 


_ 

dX  4(07-G)T 

and  ** 


8. 
4.4(*-«)T 


DIFFERENTIAL    CALCULUS. 


141 


The  curve  represented  by  this 
eouation  will  have  two  branches : 
the  one  corresponding  to  the  plus 
sign  will  be  concave  towards  the 
axis  of  abscissas,  and  the  one  cor- 
responding to  the  minus  sign  will  be 
convex.  Every  value  of  x  less  than 
a  will  render  y  imaginary.  The  co-ordinates  of  the  point 
M,  are  x  =  a,  y  —  b. 

146.     6th.    When   c  is  positive  or  negative  and  m  a 
fraction  having  an  uneven  numerator  and  an  uneven  de- 
nominator, as  m= — . 
5 

Under  this  supposition  the  equation  will  become 

£ 
j  \  /     * 

and  by  differentiating,  we  have 

dy  _  3c 

dx-~7, T' 


3.2c 


5.5(x-a)5 

from  which  we  see  that  if  we  use  the  superior  sign  of  the 
first  equation,  the  curve  will  be  convex  towards  the  axis 
of  abscissas  for  x  <  a,  that  there  will  be  a  point  of  inflexion 
for  x  —  a,  and  that  the  curve  will  be  concave  for  x  >  a. 
If  the  lower  sign  be  employed,  the  first  branch  will  become 
concave,  and  the  other  convex. 

147.    The  cusps,   which  have  been  -considered,  were 
formed  by  the  union  of  two  curves  that  were  convex  to- 


142 


ELEMENTS    OF    THE 


wards  each  other,  and  such  are  called,  cusps  of  the  first 
order. 

It  frequently  happens,  however,  that  the  curves  which 
unite,  embrace  each  other.     The  equation 


furnishes   an  example   of  this   kind.     By  extracting  the 
square  root  of  both  members  and  transposing,  we  have 


and  by  differentiating 


dx 


2 


We  see  by  examining 
the  equations,  that  the  curve 
has  two  branches,  both  of 
which  pass  through  the 
origin  of  co-ordinates.  The 
upper  branch,  which  corres- 
ponds to  the  plus  sign,  is  constantly  convex  towards  the 

axis  of  abscissas,  while  the  lower  branch  is  convex  for 
.    64  64 


\ 


,  , 

and  concave  for 


and    x  <  1 .     At 


s  225'  225 

the  last  point  the  curve  passes  below  the  axis  of  abscissas 

and  becomes  convex  towards  it.  If  we  make  the  first  dif 
ferential  coefficient  equal  to  0,  we  shall  find  x  =  0,  and 
substituting  this  value  in  the  equation  of  the  curve,  gives 
y  =  0  ;  and  hence,  the  axis  of  abscissas  is  tangent  to  both 
branches  of  the  curve  at  the  origin  of  co-ordinates.  Al 
this  point  the  differential  coefficient  of  the  second  ordei 
is  positive  for  both  branches  of  the  curve,  hence  the\ 


DIFFERENTIAL    CALCULUS. 


143 


are  both  convex  towards  the  axis.  When  the  cusp  is 
formed  by  the  union  of  two  curves  which,  at  the  point 
of  contact,  lie  on  the  same  side  of  the  common  tangent,  it 
is  called  a  cusp  of  the  second  order. 

148.  Let  us,  as  another  example,  discuss  the  curve 
whose  equation  is 

y  =  b±(x  —  a)  *Jx  —  c. 

By  differentiating,  we  obtain 


x  —  a 


We  see,  from  the  equa- 
tion of  the  curve,  that  y  will 
be  imaginary  for  all  values 
ol  x  less  than  c. 

For  x=c,  we  have  y=b; 
and  for  x  >  c,  we  have  two 
values  of  y  and  conse- 
quently two  branches  of 
the  curve,  until  x  =  a  when  they  unite  at  the  point  M. 
For  x  >  a  there  will  be  two  real  values  of  y  and  conse- 
quently two  branches  of  the  curve.  The  point  M,  at 
which  the  branches  intersect  each  other,  is  called  a  mul- 
tiple point,  and  differs  from  a  cusp  by  being  a  point 
of  intersection  instead  of  a  point  of  tangency.  At  the 
multiple  point  M  there  are  two  tangents,  one  to  each 
branch  of  the  curve.  The  one  makes  an  angle  with  the 
axis  of  abscissas,  whose  tangent  is 


10 


144  ELEMENTS    OF    THE 

the  other,  an  angle  whose  tangent  is 


149.  Besides  the  cusps  and  multiple  points  which  have 
already  been  discussed,  there  are  sometimes  other  points 
lying  entirely  without  the  curve,  and  having  no  connexion 
with  it,  excepting  that  their  co-ordinates  will  satisfy  the 
equation  of  the  curve. 

For  example,  the  equation 

ay2  —  x3  +  feo:2  =  0, 

will  be  satisfied  for  the  values 
a?  =  ±  0,  y=±Q-,  and  hence, 
the  origin  of  co-ordinates  A, 
satisfies  the  equation  of  the 
curve,  and  enjoys  the  property 
of  a  multiple  point,  since  it  is 
the  point  of  union  of  two  values 
of  a?,  and  two  values  of  y. 

If  we  resolve  the  equation  with  respect  to  y,  we  find 


y  = 


and  hence,  y  will  be  imaginary  for  all  negative  values  of 
a?,  and  for  all  positive  values  between  the  limits  x  =  0  and 
x  =  b.  For  all  positive  values  of  x  greater  than  b,  the 
values  of  y  will  be  real. 

The  first  differential  coefficient  is 


dy_ 


DIFFERENTIAL    CALCULUS.  145 

or  by  dividing  by  the  common  factor  a?, 

dy_       3x  —  2b 
dx       2Va(x  —  b) 

and  making  x  =  0,  there  results 

dy_  2b 

~   ~ 


which  is  imaginary,  as  it  should  be,  since  there  is  no  poir. 
of  the  curve  which  is  consecutive  with  the  isolated  or  con- 
jugate point.  The  differential  coefficients  of  the  higher 

orders  are  also  imaginary  at  the  conjugate  points. 

*  '%  "" 

150.     We  may  draw  the  following  conclusion  s^from  the 
preceding  discussion. 

1st.  The  equation    -j£  =  0,    determines   the    points   at 
which  the  tangents  are  parallel  to  the  axis  of  abscissas. 

2d.  The  equation      -~  —  oo  ,     determines  the  points  of 
u>x 

the  curve  at  which  the  tangents  are  perpendicular  to  the 
axis  of  abscissas.  The  two  last  equations  also  determine 
the  cusps,  if  there  are  any,  in  all  cases  where  the 
tangent  at  the  cusps  is  parallel  or  perpendicular  to  the 
axis  of  abscissas. 

3d.    The  equation      T2  =  Qy    or     ~J    =  °°    determines 
the  points  of  inflexion. 


4th.    The  equation    -j-  =     an  imaginary  constant,  i 
dec 

dicates  a  conjugate  point. 


146 


ELEMENTS    OP   THE 


CHAPTER  VII. 


Of  Osculatory  Curves — Of  Evoiutes. 


151.  Let  PT  be  tangent  to  the  curve  ABP  at  the  point 
P,  and  PN  a  normal  at  the  same  point :  then  will  PT 
be  tangent  to  the  circumference  of  every  circle  passing 
through  P,  and  having  its  centre  in  the 'normal  PN. 

It  is  plain  that  the  cen- 
tre of  a  circle  may  be 
taken  at  some  point  C, 
so  near  to  P,  that  the  cir- 
cumference shall  fall  with- 
in the  curve  APB,  and 
then  every  circumference 
described  with  a  less  ra- 
dius, will  fall  entirely 
within  the  curve.  It  is 

also  apparent,  that  the  centre  may  be  taken  at  some  point 
C',  so  remote  from  P,  that  the  circumference  shall  fall 
between  the  curve  APB  and  the  tangent  PT,  and  then 
every  circumference  described  with  a  greater  radius  will 
fall  without  the  curve.  Hence,  there  are  two  classes  of 
tangent  circles  which  may  be  described;  the  one  lying 
within  the  curve,  and  the  other  without  it. 


DIFFERENTIAL    CALCULUS. 


147 


ACE 


152.  Let   there   be 
three     curves,    APB, 
CPD,    EPF,    which 
have   a  common   tan- 
gent TP,  and  a  com- 
mon normal  PN  ;  then 
will  they  be  tangent  to 
each  other  at  the  point 
P.    It  does  not  follow, 
however,  from  this  cir- 
cumstance, that  each  curve  will  have  an  equal  tendency  to 
coincide  with  the  tangent   TP,  nor  does  it  follow  that  any 
two  of  the  curves  CPD,  EPF,  will  have  an  equal  ten 
dency  to  coincide  with  the  first  curve  APB. 

It  is  now  proposed  to  establish  the  analytical 
conditions  which  determine  the  tendency  of  curves  to 
coincide  with  each  other,  or  with  a  common  tangent. 

Designate  the  co-ordinates  of  the  first  curve  APB  by 
x  and  y,  the  co-ordinates  of  the  second  CPD  by  a/,  yf, 
and  the  co-ordinates  of  the  third  EPF  by  x",  y"  .  If  we 
designate  the  common  ordinate  PR  by  y,  y'  ',  y"  ,  we  shall 
then  have 

w,          ,    dy  h      d2y  h2       d?y      h3 


,      dy'  h      d2yf  h2 

= 


dx"  1         da/'2  1.  2      da/'3  1.  2.  3 

But  since  the  curves  are  tangent  to  each  other  at  the 
point  P,  we  have  (Art.  119), 


148  ELEMENTS    OF    THE 

*>  »d  =         =         :      h..., 


Now,  in  order  that  the  first  curve  AP.Z?  shall  approach 
more  nearly  to  the  second  CPD  than  to  the  third  EPF, 
we  must  have 


and  consequently, 


in  which  we  have  represented  the  coefficients  in  the  first 
series  by  A,  B,  C,  &c.,  and  the  coefficients  in  the  second 
by  A',  B',  a,  &c. 

Now,  the  limit  of  the  first  member  of  the  inequality  will 
always  be  less  than  the  limit  of  the  second,  when  its  first 
term  involves  a  higher  power  of  h  than  the  first  term  of 
the  second.  For,  if  A  =  0,  the  first  member  will  involve 
the  highest  power  of  h,  and  we  shall  have 


and  by  dividing  by  h2. 


and  by  passing  to  the  limit 


DIFFERENTIAL    CALCULUS  149 

'    '''*  ™ 


But  when  A  =  0,  we  have 


and  hence,  when  three  curves  have  a  common  ordinate,  the 
first  will  approach  nearer  to  the  second  than  to  the  third, 
if  the  number  of  equal  differential  coefficients  between  the 
first  and  second  is  greater  than  that  between  the  first  and 
third.  And  consequently,  if  the  first  and  second  curves 
have  m  +  1  differential  coefficients  which  are  equal  to 
each  other,  and  the  first  and  third  curves  only  m  equal  dif- 
rential  coefficients,  the  first  curve  will  approach  more 
nearly  to  the  second  than  to  the  third.  Hence  it  appears, 
that  the  order  of  contact  of  two  curves  will  depend  on 
the  number  of  corresponding  differential  coefficients  which 
are  equal  to  each  other. 

The  contact  which  results  from  an  equality  between  the 
co-ordinates  and  the  first  differential  coefficients,  is  called 
a  contact  of  the  first  order,  or  a  simple  tangency  (Art.  119). 
If  the  second  differential  coefficients  are  also  equal  to  each 
other,  it  is  called  a  contact  of  the  second  order.  If  the  first 
three  differential  coefficients  are  respectively  equal  to  each 
other,  it  is  a  contact  of  the  third  order;  and  if  there  are  m 
differential  coefficients  respectively  equal  to  each  other,  it 
is  a  contact  of  the  mth  order. 

153.  Let  us  now  suppose  that  the  second  line  is  only 
given  in  species,  and  that  values  may  be  attributed  at 
pleasure  to  the  constants  which  enter  its  equation.  We 


150  ELEMENTS    OF    THE 

shall  then  be  able  to  establish  between  the  first  and  second 
lines  as  many  conditions  as  there  are  constants  in  the 
equation  of  the  second  line.  If,  for  example,  the  equation 
of  the  second  line  contains  two  constants,  two  conditions 
can  be  established,  viz. :  an  equality  between  the  co- 
ordinates, and  an  equality  between  the  first  differential 
coefficients  ;  this  will  give  a  contact  of  the  first  order. 

If  the  equation  of  the  second  curve  contains  three  con 
stants,  three  conditions  may  be  established,  viz. :  an  equality 
between  the  co-ordinates,  and  an  equality  between  the  first 
and  second  differential  coefficients.  This  will  give  a  con- 
tact of  the  second  order.  If  there  are  four  constants,  we 
can  obtain  a  contact  of  the  third  order ;  and  if  there  are 
m  -f- 1  constants,  a  contact  of  the  mih  order. 

It  is  plain,  that  in  each  of  the  foregoing  cases  the  highest 
order  of  contact  is  determined. 

The  line  ivhich  has  a  higher  order  of  contact  with  a 
given  curve  than  can  be  found  for  any  other  line  of  the 
same  species,  is  called  an  osculatrix. 

Let  it  be  required,  for  example,  to  find  a  straight  line 
which  shall  be  oscillatory  to  a  curve,  at  a  given  point  of 
which  the  co-ordinates  are  a/f,  y" . 

The  equation  of  the  right  line  is  of  the  form 

y  =  ax  +  b, 

and  it  is  required  to  find  such  values  for  the  constants  d 
and  b  as  to  cause  the  line  to  fulfil  the  conditions, 

x  =  af',        y=y",      and  =       - 


DIFFERENTIAL    CALCULUS.  151 

By  differentiating  the  equation  of  the  line,  we  have 

dy 

~  =  a; 

ax 

and  since  the  line  passes  through  the  point  of  osculation 


Substituting  for  -~   its  value   -A/-,  we  have 

CLOC  CLCU 

y  ~y  —  if,  (x    *  )> 

for  the  equation  of  the  osculatrix. 
In  the  equation  of  the  circle 


dy  x      dy"  x" 

we  find         ~  =  --  =-/—=  --  - 

dx          y       dx"  y" 

hence,  the  equation  of  the  osculatrix  of  the  first  order,  to 
the  circle,  is 


or  by  reducing  yy"  '  -\-  xx1'  '=  R2. 

154.  If  *  and  /3  represent  the  co-ordinates  of  the  centre 
of  a  circle,  its  equation  will  be  of  the  form 


11  this  equation  be  twice  differentiated,  we  shall  have, 


J  52  ELEMENTS    OFv  THE 

and  by  combining  the  three  equations,  we  obtain, 


^ 


dxtfy 


If  it  be  now  required  to  make  this  circle  osculatory  to 
a  given  curve,  at  a  point  of  which  the  co-ordinates  are  a/', 
y;/,  we  have  only  to  substitute  in  the  three  last  equations, 
the  values  of 


<Py  _  ffy" 
dx2~dxff2' 


deduced  from  the  equation  of  the  curve,  and  to  suppose,  at 
the  same  time,  the  co-ordinates  x  and  y  in  the  curve  to 
become  equal  to  those  of  x  and  y  in  the  circle. 

If  we  suppose  x",  y"  to  beeome  general  co-ordinates 
of  the  curve,  the  circle  will  move  around  the  curve,  con- 
tinually changing  its  radius,  and  will  become  osculatory 
at  all  the  points  in  succession. 

155.  If  the  circle  CD 
be  osculatory  to  the  curve 
EF,  at  the  point  P,  we 
shall  have 

h3 


for  h  positive  ;  and 


o   E 


DIFFERENTIAL    CALCULUS.  153 

for  h  negative  :  hence,  the  two  lines  qs,  qfsr,  have  contrary 
signs.  The  curve,  therefore,  lies  above  the  oscillatory  cir- 
cle on  one  side  of  the  point  P,  and  below  it  on  the  other, 
and  consequently,  divides  the  osculatory  circle  at  the  point 
of  osculation.  Hence,  also,  the  osculatory  circle  separates 
the  tangent  circles  which  lie  without  the  curve  from  those 
which  lie  within  it  (Art.  151). 

In  every  osculatrix  of  an  even  order  the  first  term  in  the 
values  of  qs,  qfsf,  will,  in  general,  contain  an  uneven  power 
of  h  ;  and  hence  their  signs  may  be  made  to  depend  on 
that  of  h.  The  curve  will  therefore  lie  above  the  oscu- 
latrix on  one  side  of  the  point  P,  and  below  it  on  the 
other  ;  and  hence,  every  osculatrix  of  an  even  order,  will 
in  general  be  divided  by  the  curve  at  the  point  of  oscula- 

tion. 

156.  The  first  differential  equation  of  Article  154, 


may  be  placed  under  the  form 

dx, 

fs-y=-dj(x-x)- 

It  we  make  the  circle  osculatory  to  the  curve  we  have 

x  =  x",  y  =  y",     and 
dx_dx" 
~~~~''          C' 


which  is  the  equation  of  a  normal  at  the  point  whose  co- 
ordinates are  a/x  y"  (Art.  122).  But  this  normal  passes 
through  the  point  whose  co-ordinates  are  <*  and  £.  Hence, 
the  normal  drawn  through  the  point  of  osculation,  will 
contain  the  centre  of  the  osculatory  circle. 

157.  It  was  shown  in  (Art.  155.)  that  the  osculatory  cir- 
cle is,  in  general,  divided  by  the  curve  at  the  point  of  oscu- 


154  ELEMENTS    OF    THE 

lation.     The  position  of  the  curves  with  respect  to  each 
other  indicates  this  result. 

For,  the  osculatory  circle  is  always  symmetrical  with 
respect  to  the  normal,  while  the  curve  is,  in  general,  not 
symmetrical  with  respect  to  this  line.  If,  however,  the 
curve  is  symmetrical  with  respect  to  the  normal,  as  is  the 
case  in  lines  of  the  second  order  when  the  normal  coincides 
with  an  axis,  the  curve  will  not  divide  the  osculatory  circle 
at  the  point  of  osculation  ;  and  the  condition  which  renders 
the  second  differential  coefficients  in  the  curve  and  circle 
equal  to  each  other,  will  also  render  the  third  differential 
coefficients  equal,  and  the  contact  will  then  be  of  the  third 
order. 

158.  The  radius  of  the  osculatory  circle 


dx<Py 

is  affected  with  the  sign  plus  or  minus,  and  it  may  be  well 
to  determine  the  circumstances  under  which  each  sign  is 
to  be  used. 

If  we  suppose  the  ordinate  to  be  positive,  we  shall  have 
(Art.  133) 

d?y 

-y^,     and  consequently     c?y 

aoc 

negative  when  the  curve  is  concave  towards  the  axis  ot 
abscissas,  and  positive  when  it  .is  convex.  If  then,  we 
wish  the  radius  of  the  osculatory  circle  to  be  positive  for 
curves  which  are  concave  towards  the  axis  of  abscissas,  we 
must,  employ  the  minus  sign,  in  which  case  the  radius  will 
be  negative  for  curves  which  are  convex. 


DIFFERENTIAL    CALCULUS.  155 

159.  If  the  circumferences  of  two  circles  be  described 
with  different  radii,  and  a  tangent  line  be  drawn  to  each,  it 
is  plain  that  the  circumference  which  has  the  less  radius 
will  depart  more  rapidly  from  its  tangent  than  the  circum- 
ference which  is  described  with  the  greater  radius ;  and 
hence  we  say,  that  its  curvature  is  greater.     And  gener- 
ally, the  curvature  of  any  curve  is  said  to  be  greater  or  less 
than  that  of  another  curve,  according  as  its  tendency  to 
depart  from  its  tangent  at  a  given  point,  is  greater  or  less 
than  that  of  the  curve  with  which  it  is  compared. 

1 60.  The  curvature  is  the  same  at  all  the  points  of  the 
same  circumference,  and  also  in  all  circumferences  described 
with  equal  radii,  since  the  tendency  to  depart  from  the  tan- 
gent is  the  same.     In  different  circumferences,  the  curva- 
ture is  measured  by  the  angle  formed  by  two  radii  drawn 
through  the  extremities  of  an  arc  of  a  given  length. 

Let  r  and  r1  designate  the  radii  of  two  circles,  a  the 
length  of  a  given  arc  measured  on  the  circumference  of 
each ;  c  the  angle  formed  by  the  two  radii  drawn  through 
the  extremities  of  the  arc  in  the  first  circle,  and  d  the 
angle  formed  by  the  corresponding  radii  of  the  second. 
We  shall  then  have 

**r    :  a  ::  360°   :  c,     hence,     c  =  ^;  . 

also, 

360°  a 


:   a  :  :   360°   :   c/,     hence,     c'  = 
and  consequently 


156 


ELEMENTS    OF    THE 


that  is,  the  curvature  in  different  circumferences  varies 
inversely  as  the  radii. 

161.  The  curvature 
of  plane  curves  is  meas- 
ured by  means  of  the 
osculatory  circle. 

If  we  assume  two 
points  P  and  P',  either 
on  the  same  or  on  dif- 
ferent curves,  and  find 

the  radii  r  and  r1  of  the  circles  which  are  osculatory  at 
these  points,  then 

—   :   —  rj 
r        r 


curvature  at  P   :   curvature  at  Pr 


that  is,  the  curvature  at  different  points  varies  inversely 
as  the  radius  of  the  osculatory  circle. 

The  radius  of  the  osculatory  circle  is  called  the  radius 
of  curvature. 

162.  Let  us  now  determine  the  value  of  the  radius  of 
curvature  for  lines  of  the  second  order. 

The  general  equation  of  these  lines  (An.  Geom.  Bk.  VI, 
Prop.  XII,  Sch.  3),  is 


y2  =  mx  +  no?, 


which  gives, 


_  2ny  dx2—(m-}-2  nx)  dx  dy  __  [4  ny2  —  (m  +  2  nx  )2]  da? 

~~  ~~ 


DIFFERENTIAL    CALCULUS.  157 

Substituting  these  values  in  the  equation 

(dif+dyf 

dxcPy      ' 
we  obtain 

„  _  [4(7713?  +  nx2)  +  (m 


which  is  the  general  value  of  the  radius  of  curvature  in 
lines  of  the  second  order,  for  any  abscissa  x. 

163.  If  we  make  x  =  0,  we  have 
1         W 

R=^m=^' 

that  is,  in  lines  of  the  second  order,  the  radius  of  curva- 
ture at  the  vertex  of  the  transverse  axis  is  equal  to  half 
the  parameter  of  that  axis. 

If  be  required  to  find  the  value  of  the  radius  of  curva- 
ture at  the  extremity  of  the  conjugate  axis  of  an  ellipse, 
we  make  (An.  Geom.  Bk.  VIII,  Prop.  XXI,  Sch.  3), 


m  =  — — ,         n  =  —  — ,         and       x  =  A, 
A.  A. 

which  gives,  after  reducing, 


hence,  the  radius  of  curvature  at  the  vertex  of  the  conju- 
gate axis  of  an  ellipse  is  equal  to  half  the  parameter  of 
that  axis. 

In  the  case  of  the  parabola,  in  which  n  =  0,  the  genera] 
value  of  the  radius  of  curvature  becomes 


158 


ELEMENTS    OF    THE 


p 


(m2  -h 

~ 


164.  If  we  compare  the  value  of  the  radius  of  curvature 
with  that  of  the  normal  line  found  in  (Art.  118),  we  shaJ 

have 

(normal)3 


R  = 


1 


that  is,  the  radius  of  curvature  at  any  point  is  equal  to 
the  cube  of  the  normal  divided  by  half  the  parameter 
squared  :  and  hence,  the  radii  of  curvature  at  different 
points  of  the  same  curve  are  to  each  other  as  the  cubes  of 
the  corresponding  normals. 


Of  the  Evolutes  of  Curves. 


165.  If  we  suppose  an  os- 
culatory  circle  to  be  drawn  at 
each  of  the  points  of  the 
curve  APP'B,  and  then  a 
curve  ACC'C"  to  be  drawn 
through  the  centres  of  these 
circles,  this  latter  curve  is 
called  the  evolute  curve,  and 
the  curve  APP'B  the  invo- 
lute. 


166.  The  co-ordinates  of  the  centre  of  the  osculatory 
circle,  which  have  been  represented  by  <*  and  /3,  are  con- 
stant for  given  values  of  the  co-ordinates  x  and  y  of  the 


DIFFERENTIAL    CALCULUS.  159 

involute  curve,  but  they  become  variable  when  we  pass 
from  one  point  of  the  involute  curve  to  another. 

167.  We  have  already  seen  that  the  osculatory  circle  is 
characterized  by  the  equations  (Art.  154) 

\  (1) 
=  0,  (2) 
=  0.  (3) 


If  it  be  required  to  find  the  relations  between  the  co- 
ordinates of  the  involute  and  the  co-ordinates  of  the 
evolute  curves,  we  must  differentiate  equations  (1)  and  (2) 
under  the  supposition  that  <*  and  /3,  as  well  as  x  and  y, 
are  variables.  We  shall  then  have 


(x  -*)dx  +  (y-  f>)dy  -(x  —  «)d*-(y-  P)dp  =  RdR, 

da?  +  dy2  +  (y  —  P)d?y  —  d*dx  —  dftdy  =  0. 

Combining  these  with  equations  (2)  and  (3),  we  obtain 

_  (y  _  fidfi  -(x-«)d*  =  RdR,     (4) 

—  decdx  —  dftdy  =  0. 
The  last  equation  gives 


But  equation  (2)  may  be  placed  under  the  form 

dx  .          . 


which  represents  a  normal  to  the  involute  (Art.  122),  and 
which  becomes,  by  substituting  for  —  —  its  value  -^-, 


160  ELEMENTS    OF    THE 

dp .          N      t  x 
y  —  P  =  -r(x  —  *\     (6) 

or  jB  -  y  =  ^(«  -  ar)  (Art.  120). 

(Ml 

This  last  equation,  which  is  but  another  form  for  the 
equation  of  the  normal  to  the  involute,  is,  in  fact,  the 
equation  of  a  tangent  line  to  the  evolute,  at  the  point 
whose  co-ordinates  are  a  and/3;  hence,  a  normal  line  to 
the  involute  curve  is  tangent  to  the  evolute. 

168.  It  is  now  proposed  to  show,  that  the  radius  of  cur- 
vature and  the  evolute  curve  have  equal  differentials 
Combining  equations  (2)  and  (5)  we  obtain 

<*$M    ,h    (x-.)  =  (y-t)*L,  (7)    ..  .A,...  , 

or  by  squaring  both  members, 


combining  this  last  with  equation  (1)  we  have 
|jj!]    fiWJtfVtf-*.     (8) 
Combining  equations  (4)  and  (7),  we  have 


or 


DIFFERENTIAL    CALCULUS.  161 

or  by  squaring  both  members 


Dividing  this  last  by  equation  (8),  member  by  member 
we  have 


or  dR  =  Vd«*  +  dp. 

But  if  s  represents  the  arc  of  the  evolute  curve,  of  which 
the  co-ordinates  are  *  and  £,  we  shall  have  (Art.  128), 


ds= 
hence,  dR  =  ds  ; 

that  is,  the  differential  of  the  radius  of  curvature  is  equal 
to  the  differential  of  the  arc  of  the  evolute. 

169.  It  does  not  follow,  however,  from  the  last  equation, 
that  the  radius  of  curvature  is  equal  to  the  arc  of  the  evolute 
curve,  but  only  that  one  of  them  is  equal  to  the  other  plus 
or  minus  a  constant  (Art.  22).  Hence, 


is  the  form  of  the  equation  which  expresses  the  relation 
between  them. 


162 


ELEMENTS    OF    THE 


If  we  determine  the  radii 
of  curvature  at  two  points  of 
the  involute,  as  P  and  Pf, 
we  shall  have,  for  the  first, 


and  for  the  second 


hence, 


and  hence,  the  difference  between  the  radii  of  curvature  at 
any  two  points  of  the  involute  is  equal  to  the  part  of  the 
evolute  curve  intercepted  between  them. 

170.  The  value  of  the  constant  a  will  depend  on  the 
position  of  the  point  from  which  the  arc  of  the  evolute 
curve  is  estimated. 

If,  for  example,  we  take  the  radius  of  curvature  for  lines 
of  the  second  order,  and  estimate  the  arc  of  the  evolute 
curve  from  the  point  at  which  it  meets  the  axis,  the  value 
of  s  will  be  0  when  R  =  —  m  (Art.  163):  hence  we 
shall  have 


or     a  =  — i 


and  for  any  other  point  of  the  curve 

J_ 

2 


DIFFERENTIAL    CALCULUS. 


163 


Either  of  the  evolutes,  FE, 
FEf,  F'E',  or  F'E,  corres- 
ponding to  one  quarter  of  the 
ellipse,  is  equal  to  (Art.  169) 

A2         £2 
B          A   ' 

171 .  The  evolute  curve  takes 
its  name  from  the  connexion  which  it  has  with  the  corres 
ponding  involute. 

Let  CC7C/7  be  an  evolute 
curve.  At  C  draw  a  tan- 
gent AC,  and  make  it  equal 
to  the  constant  a  in  the  equa- 
tion 


Wrap  a  thread  ACC7C7/ 
around  the  curve,  and  fasten 
it  at  any  point,  as  C7/. 

Then,  if  we  begin  at  A, 
and  unwrap   or   evolve   the 
thread,  it  will  take  the  positions  PC7,  P7C77,  &c.,  and  the 
point  A  will  describe  the  involute  APP7 :  for 

PC7-AC=CC7     and     P7C/7-  AC=  CC7C77,  &c.  .  .  . 

172.  The  equation  of  the  evolute  may  be  readily  found 
by  combining  the  equations 

dy(da?+dif) 
dxcPy    ~' 


_ 


with  the  equation  of  the  involute  curve. 


164  ELEMENTS    OF    THE 

1st.  Find,  from  the  equation  of  the  involute,  the  values  of 

~     and     c^y, 
dx 

and  substitute  them  in  the  two  last  equations,  and  there 
will  be  obtained  two  new  equations  involving  *,  £,  x  and  y. 
2d.  Combine  these  equations  with  the  equation  of  the 
involute,  and  eliminate  x  and  y  :  the  resulting  equation 
will  contain  *,  /3,  and  constants,  and  will  be  the  equation 
of  the  e  volute  curve. 

173.  Let  us  take,  as  an  example,  the  common  parabola 
of  which  the  equation  is 

y2  =  77107. 

We  shall  then  have 

dy  _  m  „  m2da? 

''  "" 


and  hence 

_  4y3/4y2  +  ^2\  _  4y3  +  ^2y  _  4y 
"~5?A     4y2     )-     ~rf~      ~^ 

and  by  observing  that  the  value  of  x  —  «•  is  equal  to  that 
of  y  —  /3  multiplied  by     —  --->     we  nave 


hence  we  have, 


and    a?  —  *=  --  f  --      : 
m        2 


DIFFERENTIAL    CALCULUS-  165 

substituting  for  y  its  value  in  the  equation  of  the  involute 


y  — 


we  obtain 


m 
--; 


and  by  eliminating  a?,  we  have 


16 

27m' 


-»> 


which  is  the  equation  of  the  evolute. 
If  we  make  ft  =  0,  we  have 

J_ 

~  2  ™' 

and  hence,  the  evolute  meets  the 
axis  of  abscissas  at  a  distance  from 
the  origin  equal  to  half  the  param- 
eter. If  the  origin  of  co-ordinates 
be  transferred  from  A  to  this 
point,  we  shall  have 


and  consequently 


21m 


The  equation  of  the  curve  shows  that  it  is  symmetrical 
with  respect  to  the  axis  of  abscissas,  and  that  it  does  not 
extend  in  the  direction  of  the  negative  values  of  <*! .  The 
evolute  CCr  corresponds  to  the  part  AP  of  the  involute, 
and  CClf  to  the  part  AP' . 


166 


ELEMENTS    OF    THE 


CHAPTER   VIII. 

Of  Transcendental  Curves. — Of  Tangent  Planes 
and  Normal  Lines  to  Surfaces. 

174.  Curves  may  be  divided  into  two  general  classes  . 
1st.  Those  whose  equations  are  purely  algebraic  ;  and 
2dly.    Those   whose   equations   involve    transcendental 

quantities. 

The  first  class  are  called  algebraic  curves,  and  the 
second,  transcendental  curves. 

The  properties  of  the  first  class  having  been  already 
examined,  it  only  remains  to  discuss  the  properties  of  the 
transcendental  curves. 

Of  the  Logarithmic  Curve. 

175.  The  logarithmic  curve  takes  its  name  from  the 
property  that,  when  referred  to  rectangular  axes,  one  of 
the  co-ordinates  is  equal  to  the  logarithm  of  the  other. 

If  we  suppose  the  logarithms  to  be  estimated  in  paral- 
lels to  the  axis  of  Y,  and  the  corresponding  numbers  to 
be  laid  off  on  the  axis  of  abscissas,  the  equation  of  the 
curve  will  be 

y  =:  Ix. 


DIFFERENTIAL    CALCULUS. 


176.  If  we  designate  the 
base  of  a  system  of  loga- 
rithms by  a,  we  shall  have, 
(Alg.  Art.  241) 

ay  —  x\ 

and  if  we  change  the  value 
of  the  base  a  to  a',  we  shall 
have 

afy  =  x. 


It  is  plain,  that  the  same  value  of  a?,  in  the  two  equations, 
will  give  different  values  of  y,  and  hence,  every  system  of 
logarithms  will  give  a  different  logarithmic  curve. 

If  we  make  y  —  0,  we  shall  have  (Alg.  Art.  257) 
x  •=.  1 ;  and  this  relation  being  independent  of  the  base  of 
the  system  of  logarithms,  it  follows,  that  every  logarithmic 
curve  will  intersect  the  axis  of  numbers  at  a  distance  from 
the  origin  equal  to  unity. 

The  equation 


ay  =  a?. 


will  enable  us  to  describe  the  curve  by  points,  even  with- 
out the  aid  of  a  table  of  logarithms.     For,  if  we  make 


we  shall  find,  for  the  corresponding  values  of  x, 

x  =  a  -y/a,         x  =  <\/a     &c. 


a?  — 


177.  If  we  suppose  the  base  of  the  system  of  logarithms 
to  be  greater  than  unity,  the  logarithms  of  all  numbers  less 


168         |  ELEMENTS    OF    THE 

than  unity  will  be  negative  ( Alg.  Art.  256) ;  and  therefore, 
the  values  of  y  corresponding  to  the  abscissas,  between  the 
limits  a?  — 0  and  x  =  AE  =  l,  will  be  negative.  Hence, 
these  ordinates  are  laid  off  below  the  axis  of  abscissas. 

When  x  —  0,y  will  be  infinite  and  negative  (Alg.  Art. 
264).  If  we  make  x  negative,  the  conditions  of  the  equa- 
tion cannot  be  fulfilled ;  and  hence,  the  curve  does  not 
extend  on  the  side  of  the  negative  abscissas. 

178.  Let  us  resume  the  equation  of  the  curve 

y  =  Ix. 

11  we  represent  the  modulus  of  the  system  of  logarithms 
by  Aj  and  differentiate,  we  obtain  (Art.  56), 

,         Adx 
dy  =  A-, 

dy      A 

or  •/-  =  —' 

ax      x 

But   —    represents  the  tangent  of  the  angle  which  the 

CLCC 

tangent  line  forms  with  the  axis  of  abscissas  :  hence,  the 
tangent  will  be  parallel  to  the  axis  of  abscissas  when 
x  —  oo  ,  and  perpendicular  to  it  when  x  =  0. 

But  when  x  =  0,  y  —  —  oo  ;  hence,  the  axis  of  ordinates 
is  an  asymptote  to  the  curve.  The  tangent  which  is 
parallel  to  the  axis  of  X  is  not  an  asymptote :  for  when 
x  —  oo  ,  we  also  have  y  —  oo  . 

179.  The  most  remarkable  property  of  this  curve  be 
longs  to  its  sub-tangent   FR',   estimated  on  the  axis  of 
logarithms.     We  have  found,  for  the  sub-tangent,  on  the 
axis  of  X  (Art.  114), 


DIFFERENTIAL    CALCULUS. 


169 


and  by  simply  changing  the  axes,  we  have 


dx 

hence,  the  sub-tangent  is  equal  to  the  modulus  of  the 
system  of  logarithms  from  which  the  curve  is  constructed. 
In  the  Naperian  system  M  =  l,  and  hence  the  sub-tangent 
will  be  equal  to  1  =  AE. 


Of  the  Cycloid. 


G  , 


B 


A  N 

180.  If  a  circle  NPG  be  rolled  along  a  straight  line 
AL,  any  point  of  the  circumference  will  describe  a  curve, 
which  is  called  a  cycloid.     The  circle  NPG  is  called  the 
generating  circle,  and  P  the  generating  point. 

It  is  plain,  that  in  each  revolution  of  the  generating  circle 
an  equal  curve  will  be  described ;  and  hence,  it  will  only 
be  necessary  to  examine  the  properties  of  the  curve 
APBL,  described  in  one  revolution  of  the  generating  circle. 
We  shall  therefore  refer  only  to  this  part  when  speaking 
of  the  cycloid. 

181.  If  we  suppose  the- point  P  to  be  on  the  line  AL 
at  Ay  it  will  oe  found  at  some  point,  as  L9  after  all  the 


170 


ELEMENTS    OF    THE 


A    R 


N 


M 


points  of  the  circumference  shall  have  been  brought  in 
contact  with  the  line  AL.  The  line  AL  will  be  equal  to 
the  circumference  of  the  generating  circle,  and  is  called 
the  base  of  the  cycloid.  The  line  J3M,  drawn  perpen 
dicular  to  the  base  at  the  middle  point,  is  equal  to  the 
diameter  of  the  generating  circle,  and  is  called  the  axis  of 
the  cycloid. 

182.  To  find  the  equation  of  the  cycloid,  let  us  assume 
the  point  A  as  the  origin  of  co-ordinates,  and  let  us  sup- 
pose that  the-  generating  point  has  described  the  arc  A  P. 
If  N  designates  the  point  at  which  the  generating  circle 
touches  the  base,  AN  will  be  equal  to  the  arc  NP. 

Through  N  draw  the  diameter  NG,  which  will  be 
perpendicular  to  the  base.  Through  P  draw  PR  perpen- 
dicular to  the  base,  and  PQ  parallel  to  it.  Then,  PR  =  NQ 
will  be  the  versed-sine,  and  PQ  the  sine  of  the  arc  NP. 

Let  us  make 


ON  =  r, 

we  shall  then  have 


=  NQ=y, 


-y*,       x  =  AN-RN=a.icNP-PQ: 

hence,  the  transcendental  equation  is 


x  =  ver-sin"1^  —  V%  ry  —  y2. 


DIFFERENTIAL    CALCULUS.  171 

183.  The  properties  of  the  cycloid  are,  however,  most 
easily  deduced  from  its  differential  equation,  which  is 
readily  found  by  differentiating  both  members  of  the  trans- 
scendental  equation. 

We  have  (Art.  71), 

r/(     '  -1  ^-     rdy 

V~2~ry-y2' 
rdy  —  ydy 

V2ry-y2' 
hence, 

^.          rd!/ 


or 

—  y2 

which  is  the  differential  equation  of  the  cycloid. 

184.  If  we  substitute  in  the  general  equations  of  (Arts. 
114,  115,  116,  117),  the  values  of  dx,  dy>  deduced  from 
the  differential  equation  of  the  cycloid,  we  shall  obtain  the 
values  of  the  normal,  sub-normal,  tangent,  and  sub-tangent. 
They  are, 

normal  PN  =  V%ry,         sub-normal  RN  =  V^ry  —  y2, 


sub-tangent  TR= 


These  values  are  easily  constructed,  in  consequence  of 
their  connexion  with  the  parts  of  the  generating  circle. 

The  sub-normal  RN,  for  example,  is  equal  to  PQ  of 
the  generating  circle,  since  each  is  equal  to  -y/2ry  —  y2  : 
hence,  the  normal  PN  and  the  diameter  GN  intersect 
the  base  of  the  cycloid  at  the  same  point. 


172  ELEMENTS    OP    THE 

Now,  since  the  tangent  to  the  cycloid  at  the  point  P  is 
perpendicular  to  the  normal,  it  must  coincide  with  the 
chord  PG  of  the  generating  circle. 

If,  therefore,  it  be  required  to  draw  a  normal  or  a  tan- 
gent to  the  cycloid,  at  any  point  as  P,  draw  any  line,  as 
ng,  perpendicular  to  the  base  AL,  and  make  it  equal  to 
the  diameter  of  the  generating  circle.  On  ng  describe  a 
semi-circumference,  and  through  P  draw  a  parallel  to  the 
base  of  the  cycloid.  Through  p,  where  the  parallel  cuts 
the  semi-circumference,  draw  the  supplementary  chords 
pn,  pg,  and  then  draw  through  P  the  parallels  PN,  PG, 
and  PN  will  be  a  normal,  and  PG  a,  tangent  to  the  cycloid 
at  the  point  P. 

185.  Let  us  resume  the  differential  equation  of  the 
cycloid 


which  may  be  put  under  the  form 


dx  y  y 

If  we  make  y  =  0,  we  shall  have 

dy_ 

dx~ 

and  if  we  make  y  =  2r,  we  shall  have 


DIFFERENTIAL    CALCULUS.  173 

hence,  the  tangent  lines  drawn  to  the  cycloid  at  the  points 
where  the  curve  meets  the  base,  are  perpendicular  to  the 
base;  and  the  tangent  drawn  through  the  extremity  of  the 
greatest  ordinate,  is  parallel  to  the  base. 
186.  If  we  differentiate  the  equation 


=, 

V2ry  —  yz 
regarding  dx  as  qonstant,  we  obtain 


or  by  reducing  and  dividing  by  y, 


whence  we  obtain 


and  hence  the  cycloid  is  concave  towards   the  axis  of 
abscissas  (Art.  133). 

187.  To  find  the  evolute  of  the  cycloid,  let  us  first  sub- 
stitute in  the  general  value  of 


dxtfy 


the  value  of  d*y  found  in  the  last  article  :  we  shall  then 
have 


hence,  the  radius  of  curvature  corresponding  to  the  ex- 
tremity of  any  ordinate  y,  is  equal  to  double  the  normal. 


174 


ELEMENTS    OF    THE 


The  radius  of  curvature  is  0  when  y  =  0,  and  equal  tc 
twice  the  diameter  of  the  generating  circle  for  y  =  2r: 
hence,  the  length  of  the  e volute  curve  from  A  to  A'  is 
equal  to  twice  the  diameter  of  the  generating  circle. 

Substituting  the  value  of  cPy  in  the  values  of  y—  /3, 
x—  *  (Art.  172),  we  obtain 


y  —  P  = 
hence  we  have 


x  —  *  —  2  V—  2r/3  —  /32. 


Substituting  these  values  of  y  and  x  in  the  transcen- 
dental equation  of  the  cycloid,  we  have 


which  is  the  transcendental  equation  of  the  evolute,  re- 
ferred to  the  primitive  origin  and  the  primitive  axes. 

Let  us  now  trans- 
fer the  origin  of  co- 
ordinates to  the  point 
A',  and  change  at 
the  same  time  the 
direction  of  the  posi- 
tive abscissas :  that 
is,  instead  of  estima- 
ting them  from  the 
left  to  the  right,  we  will  estimate  them  from  the  rigi 
to  the  left.  Let  us  designate  the  co-ordinates  of  tl-j 
evolute,  referred  to  the  new  axes  A!  M,  AfXf,  by  *'  and  f.f 


DIFFERENTIAL    CALCULUS.  175 

Since  A'X'  =  AM  =  the  semi-circumference  of  the  gene- 
rating circle,  which  is  equal  to  m,  we  shall  have,  for  the 
abscissa  ArRr  of  any  point  Px, 

AfRf  —ct!  =  rir  —  *,      hence,      *  =  r*  — •  «f : 
and  for  the  ordinate,  we  shall  have 

RfP'=  p'  =  R'E  -  P'E  =  2r-  (-  /3)  =  2r  +  ft, 
hence,         ft  =  —  2r  +  ft',       or       —  ft  =  2r  —  /a'. 

Substituting  these  values  of  *  and  ft  in  the  transcen 
dental  equation  of  the  evolute,  we  obtain 


TV  —  a!  —  ver-sin"1  (2r  —  ftf)  - 

or        af=r*  —  ver-sin"1  (2  r  —  ft')  —  V2  rfi'  —  ft'2. 

But  the  arc  whose  versed-sine  is  2r  —  ftf,  is  the  supple 
ment  of  the  arc  whose  versed-sine  is  ft',  hence 


a!  —  ver-sin  -1    &  —  -\/2rpf—  p'2, 

which  is  the  equation  of  the  evolute  referred  to  the  new 
origin  and  new  axes. 

But  this  equation  is  of  the  same  form,  and  involves  the 
same  constants  as  that  of  the  involute :  hence,  the  evolute 
and  involute  are  equal  curves. 

Of  Spirals. 

188.  A  spiral  is  a  curve  described  by  a  point  which 
moves  along  a  right  line,  according  to  any  law  whatever, 

the  line  having  at  the  same  time  a  uniform  angular  motion. 

12 


176 


ELEMENTS    OP    THE 


Let  A B  C  be  a  straight 
line  which  is  to  be  turned 
uniformly  around  the 
point  A.  When  the 
motion  of  the  line  be- 
gins, let  us  suppose  a 
point  to  move  from  A 
along  the  line  in  the 
direction  ABC.  When 
the  line  takes  the  posi- 
tion ADE  the  point  will 

have  moved  along  it  to  some  point  as  D,  and  will  have 
described  the  arc  AaD  of  the  spiral.  When  the  line 
takes  the  position  AD'Er  the  point  will  have  described 
the  curve  AaDDf,  and  when  the  line  shall  have  comple- 
ted an  entire  revolution  the  point  will  have  described  the 
curve  AaDD'B. 

The  point  A,  about  which  the  right  line  moves,  is 
called  the  pole  ;  the  distances  AD,  AD',  AB,  are  called 
radius-vectors,  and  if  the  revolutions  of  the  radius-vector 
are  continued,  the  generating  point  will  describe  an  in- 
definite spiral.  The  parts  AaDD'B,  BFF'C,  described  in 
each  revolution,  are  called  spires. 

189.  If  with  the  pole  as  a  centre,  and  AB,  the  distance 
passed  over  by  the  generating  point  in  the  direction  of  the 
radius-vector  during  the  first  revolution,  as  a  radius,  we 
describe  the  circumference  BEE',  the  angular  motion  of 
the  radius-vector  about  the  pole  A,  may  be  measured  by 
the  arcs  ot  this  circle,  estimated  from  B. 

If  we  designate  the  radius-vector  by  u,  and  the  measur- 
ing arc,  estimated  from  B,  by  t,  the  relation  between  u 


X 

DIFFERENTIAL    CALCULUS.  177 

and  t,  may  in  general  be  expressed  by  the  equation 
u  =  atn, 

in  which  n  depends  on  the  law  according  to  which  the 
generating  point  moves  along  the  radius-vector,  and  a  on 
the  relation  which  exists  between  a  given  value  of  u  and 
the  corresponding  value  of  t. 

190.  When  n  is  positive  the  spirals  represented  by  the 
equation 

u  =  atn, 

will  pass  through  the  pole  A.  For,  if  we  make  t  ==  0,  we 
shall  have  u  —  0. 

But  if  n  is  negative,  the  equation  will  become 


,         or        U=JT' 
in  which  we  shall  have 

u  =  CD         for        t  =  0, 
and  u  =  0          for         t=co: 

hence,  in  this  class  of  spirals,  the  first  position  of  the 
generating  point  is  at  an  infinite  distance  from  the  pole  : 
the  point  will  then  approach  the  pole  as  the  radius-vector 
revolves,  and  will  only  reach  it  after  an  infinite  number  of 
revolutions. 

191.  If  we  make  n  =  1,  the  equation  of  the  spiral  be- 
comes 

u  =  at. 

If  we  designate  two  different  radius-vectors  by  u1  and 
u",  and  the  corresponding  arcs  by  tf  and  if',  we  shall  have 

u'  =  atf,        and        u"  =  at", 


178 


ELEMENTS    OF    THE 


and  consequently 


.  .    tr    . 
.  .    i     . 


that  is,  the  radius-vectors  are  proportional  to  the  measur- 
ing arcs,  estimated  from  the  point  B.  This  spiral  is 
called,  the  spiral  of  Archimedes. 

192.  If  we  represent  by  unity  the  distance  which  the 
generating  point  moves  along  the  radius-vector,  during  one 
revolution,  the  equation 

u  =  at, 
will  become 


1  —at, 


or 


£*» 

a 


a  = 


But   since   t  is  the  circumference   of  a   circle  whose 

radius  is  unity,  we  shall  have 

j  i 

—  =  2*-,         and  consequently, 

193.  If  the  axis  BD,  of 
a  semi-parabola  BCD,  be 
wrapped  around  the  circum- 
ference of  a  circle  of  a 
given  radius  r,  any  abscissa, 
as  Bb,  will  coincide  with 
an  equal  arc  Bbf,  and  any 
ordinate  as  ba,  will  take  the 
direction  of  the  normal  Ab'af. 
The  curve  Ba'cf,  described 

through  the  extremities  of  the  ordinates  of  the  parabola,  is 
called  the  parabolic  spiral. 

The  equation  of  this  spiral  is  readily  found,  by  observing 
that  the  squares  of  the  lines  b'a',  c  cf,  &c.,  are  propor- 
tional to  the  abscissas  or  arcs  Bbf,  Be  . 


DIFFERENTIAL    CALCULUS.  179 

If  we  designate  the  distances,  estimated  from  the  pole 
y  by  u,  we  shall  have  Va!  =  u  —  r:  hence, 


is  the  equation  of  the  parabolic  spiral. 

If  we  suppose  r  =  0,  the  equation  becomes 

u2  =  2pt. 

If  we  make  n  —  —  1  ,  the  general  equation  of  spirals 
becomes 

u  =  at~l,         or         ut  =  a. 

This  spiral  is  called  the  hyperbolic  spiral,  because  of  the 
analogy  which  its  equation  bears  to  that  of  the  hyperbola, 
when  referred  to  its  asymptotes. 

194.  The  relation  between  u  and  t  is  entirely  arbitrary, 
and  besides  the  relations  expressed  by  the  equation 


we  may,  if  we  please,  make 

t  =  \ogu. 

The  spiral  described  by  the  extremity  of  the  radius-vec- 
tor when  this  relation  subsists,  is  called  the  logarithmic 
spiral. 

195.  If  in  the  equation  of  the  hyperbolic  spiral,  we 
make  successively, 

1  1  1 

t=l>       =  ?      =  3'      =4'&C" 

we  shall  have  the  corresponding  values, 

u  =  a,     u  =  2a,     u  =  3a,     u  —  4a,  &c. 


180 


ELEMENTS    OF    THE 


Through  the 
pole  A  draw  AD 
perpendicular  to 
ABy  and  make 
it  equal  to  a : 
then  through  D 
draw  a  parallel 
to  AB.  From 
any  point  of  the 

spiral   as  P  draw  PM  perpendicular  to   AB,  we   shall 
then  have 

PM  =  u  sin  MAP  =  u  sin  t. 


If  we  substitute  for  u  its  value  — ,  we  shall  have 


PM=a 


sin* 


smt 


Now  as  the  arc  t  diminishes,  the  ratio  of will  ap- 
proach to  unity,  and  the  value  of  the  ordinate  PM  will 
approach  to  a  or  CM:  hence,  the  line  DC  approaches 
the  curve  and  becomes  tangent  to  it  when  t  =  0.  But 
when  t  =  0,  u  =  oo  ;  hence,  the  line  DC  is  an  asymptote 
of  the  curve. 

196.  The  arc  which  measures  the  angular  motion  of  the 
radius-vector  has  been  estimated  from  the  right  to  the  left, 
and  the  value  of  t  regarded  as  positive.  If  we  i evolve 
the  radius-vector  in  a  contrary  direction,  the  measuring 
arc  will  be  estimated  from  left  to  right,  the  sign  of  t  will 
be  changed  to  negative  and  a  similar  spiral  will  be  de- 
scribed. The  line  DO  is  an  asymptote  to  the  hyperbolic 
spiral,  corresponding  to  the  negative  value  of  t. 


DIFFERENTIAL    CALCULUS. 


181 


197.  Let  us  now  find  a  general  value  for  the  subtan- 
gent of  any  curve  referred  to  polar  co-ordinates.  The 
subtangent  is  the  projection  of  the  tangent  on  a  line 
drown  through  the  pole  and  perpendicular  to  the  radius- 
vector  passing  through  the  point  of  contact. 

The  equation  of  the  curve  may  be    written  under   the 

*brrn 

u=f(t\ 

in  which  we  may  suppose  t  the  independent  variable,  and 
its  first  differential  constant. 

Let  A  O  =  1  be  the  radius  of 
the  measuring  circle,  P  T  a  tan- 
gent to  the  curve  at  the  point  P, 
and  A  T  drawn  perpendicular  to 
the  radius-vector  AP,  the  sub- 
tangent. 

Take  any  other  point  of  the 
curve  as  P',  and  draw  APf. 
About  the  centre  A  describe  the 
arc  PQ,  and  draw  the  chord  PQ. 
Draw  also  the  secant  PPr  and 
prolong  it  until  it  meets  AT, 
drawn  parallel  to  QP,  at  T. 

From  the  similar  triangles    QPP',  A  T'P'y  we  have 


hence, 


PQ   :    QP'   ::   AT'   :  AP' '; 

QP     AP1 
PQ  "  AT1' 


But  when  we  pass  to  the  limit,  by  supposing  the  point 
P1  to  coincide  with  P,  the  secant  TPP'  will  become  the 
tangent  PT,  and  AT  will  become  the  subtangent  AT. 


182 


ELEMENTS    OF    THE 


But  under  this  supposition 
the  arc  NN'  will  become  equal 
to  dt,  the  arc  PQ  to  the  chord 
PQ  (Art.  128),  AP'  to  u,  and 
the  line  QPf  to  du. 

To  find  the  value  of  the  arc   I 
PQ,  wre  have 

1    :   NN1  :  :  AP  :   arc  PQ  ; 
hence, 

1    :   dt  :  :   u   :   arc  PQ, 
and  PQ  =  udt. 

Substituting  these  values,  and  passing  to  the  limit,  we 
have 

du         u 

~udi~~~AT: 

hence,  we  have  the  subtangent 

U2dt 

'  du  ' 

198.  If  we  find  the  value  of  u2  and  du  from  the  gen- 
eral equation  of  the  spirals 

we  shall  have 


AT=^ 


DIFFERENTIAL    CALCULUS.  183 

In  the  spiral  of  Archimedes,  we  have 

n  =  I,       and       a=  —  ; 
£K 

t2 

hence,  AT—  —  . 

2-r 

If  now  we  make  t  =  %ir=  circumference  of  the  mea- 
suring circle,  we  shall  have 

A  T  —  27f  —  circumference  of  measuring  circle. 
After  7?i  revolutions,  we  shall  have 


and  consequently, 

A  T  =  2  m2*  =  m  .  2  m*  ; 


that  is,  the  subtangent,  after  m  revolutions,  is  equal  to 
m  times  the  circumference  of  the  circle  described  with 
the  radius-vector.  This  property  was  discovered  by 
Archimedes. 

199.  In  the  hyperbolic  spiral  n  =  —  1,  and  the  value  of 
the  subtangent  becomes 

AT=-a; 

that  is,  the  subtangent  is  constant  in  the  hyperbolic  spiral. 

200.  It  may  be  remarked,  that 

AT  _udt 
AP  ~  du 

expresses  the  tangent  of  the  angle  which  the  tangent  makes 
with  the  radius-vector 


184  ELEMENTS    OF   THE 

In  the  logarithmic  spiral,  of  which  the  equation  is 


we  have  dt  =  A  —  ; 

AT     udt 
hence,  _=__=A; 

that  is,  in  the  logarithmic  spiral,  the  angle  formed  by  the 
tangent  and  the  radius-vector  passing  through  the  point  of 
contact,  is  constant  ;  and  the  tangent  of  the  angle  is  equal 
to  the  modulus  of  the  system  of  logarithms.  If  t  is  the 
Naperian  logarithm  of  u,  the  angle  will  be  equal  to  45°. 

201.  The  value  of  the  tangent  in  a   curve   referred  to 
polar  co-ordinates, 


du*' 

202.  To  find  the  differential  of  the  arc,  which  we  will 
represent  by  z,  we  have 


or,  by  substituting  for    QPf   and  PQ   their  values,  and 
passing  to  the  limit,  we  have 


DIFFERENTIAL    CALCULUS. 


185 


203.  The  differential  of  the 
area  ADP  when  referred  to  the 
Dolar  co-ordinates,  is  not  an  ele- 
mentary rectangle  as  when  re- 
ferred to  rectangular  axes,  but 
is  the  elementary  sector  APPf. 

The  limit  of  the  ratio  of  the 
sector  APP'  with  the  arc  NN', 
will  be  the  same  as  that  of 
either  of  the  sectors  APQ, 
A.P"P'  between  which  it  is 
contained,  with  the  same  arc 
NN'.  Hence,  if  we  designate 
the  area  by  s,  and  pass  to  the  limit,  we  shall  have 

APxPQ      u2  u2dt 


ds 
dt 


2NNf 


=  —       or 


which  is  the  differential  of  the  area  of  any  segment  ol  a 
spiral. 


Of  Tangent  Planes  and  Normal  Lines  to  Surfaces. 

204.  Let  u  =  F(x,y,z)  =  0, 

be  the  equation  of  a  surface. 

If  through  any  point  of  the  surface  two  planes  be  passed 
intersecting  the  surface  in  two  curves,  and  two  straight 
lines  be  drawn  respectively  tangent  to  each  of  the  curves, 
at  their  common  point,  the  plane  -of  these  tangents  will  be 
tangent  to  the  surface. 

205.  Let  us  designate  the  co-ordinates  of  the  point  at 
which  the  plane  is  to  be  tangent  by  a/',  y",  zff. 


186  ELEMENTS    OF    THE 

Through  this  point  let  a  plane  be  passed  parallel  to  the 
co-otdinate  plane  YZ.  This  plane  will  intersect  the 
surface  in  a  curve.  The  equations  of  a  straight  line  tan- 
gent to  this  curve,  at  the  point  whose  co-ordinates  are 
a/',y",zr',  are 

r   _    rJI    _  fjff  ..  .//    —    ^y_(y^    y/f\   . 

3  >      y  —  y  ~~fo?\zi?  /» 

the  first  equation  represents  the  projection  of  the  tangent 
on  the  co-ordinate  plane  ZX,  and  the  second  its  projec- 
tion on  the  co-ordinate  plane  YZ  (An.  Geom.  Bk.  IX 
Art.  70). 

Through  the  same  point  let  a  plane  be  passed  parallel  to 
the  co-ordinate  plane  ZX,  and  we  shall  have  for  the 
equations  of  a  tangent  to  the  curve 


The  coefficient    -j-     represents  the  tangent  of  the  angle 

which  the  projection  of  the  first  tangent  on  the  co-ordinate 
plane  YZ  makes  with  the  axis  of  Z  ;  and  the  coefficient 
j-  represents  the  tangent  of  the  angle  which  the  projection 

of  the  second  tangent  on  the  plane  ZX  makes  with  the 
axis  of  Z  (An.  Geom.  Bk.  VIII,  Prop.  II). 

But  these  coefficients  may  be  expressed  in  functions  of 
the  surface  and  the  co-ordinates  of  its  points.     For,  we 

have 

u  =  f(x,y,z)  =0, 

and  if  we  suppose  x  constant,  we  shall  have  (Art.  87) 

,        du  j     .  du  j 
du  =  —dy  +  —dz  =  Q: 
dy  *      dz 


DIFFERENTIAL    CALCULUS.  187 

du 


hence,  -r~=  —  7-  : 

dz          du 


., 

and  if  we  suppose  y  constant,  we  shall  find,  in  a  similar 
manner, 

du 

dx  _        dz 

dz  du 

~dx 

hence,  the  equation  of  the  projection  of  the  first  tangent  on 
the  plane  of  YZ  becomes 


dy 

and  the  equation  of  the  projection  of  the  Second  tangent 
on  the  plane  of  ZX  is 

du 

x-a/'=:--^-(z-z"). 
du  v 

dx 

The  equation  of  a  plane  passing  through  the  point  whose 
co-ordinates  are  a/x,  y"  ,  z"  is  of  the  form 


in  which  -—will  represent  the  tangent  of  the  angle  which 
the  trace  on  the  co-ordinate  plane   YZ  makes  with  the 

£f 

axis  of  Z,  and  —  ~r^Q  tangent  of  the  angle  which  the 
A. 

trace  on  the  plane  of  ZX.  makes  with  the  axis  of  Z. 


188  ELEMENTS    OP    THE 

But  since  the  tangents  are  respectively  parallel  to  the 
co-ordinate  planes  YZ,  ZX,  their  projections  will  be 
parallel  to  the  traces  of  the  tangent  plane  :  therefore, 

du  du 


B 

~~fa' 

nence,  —  £>  =  5*-  L/  ; 

a* 

dy 

dz 

du 

du 

C 

A  ~ 

dz 
~  du' 

hence,  —  A  =  r—  C. 
rfw 

Tx 

22 

Substituting  these  values  of  B  and  A  in  the  equation 
of  the  plane,  and  reducing,  we  find 

...du      ,          ,/\du      ,          ff.du 
(z^zn}_  +  (x_y/,}_  +  (y_y:l)_^y 

which  is  the  equation  of  a  tangent  plane  to  a  surface  at  a 
point  of  which  the  co-ordinates  are   a/',  yv,  z"  . 

206.  A  normal  line  to  the  surface  being  perpendicular 
to  the  tangent  plane  at  the  point  of  contact,  its  equations 
will  be  of  the  form 

du  du 


dz  dz 


ELEMENTS 


OF   THE 


INTEGRAL     CALCULUS, 


Integration  of  Differential  Monomials. 

207.  The  Differential  Calculus  explains  the  method  of 
finding  the  differential  of  a  given  function.     The  Integral 
Calculus  is  the  reverse  of  this.     It  explains  the  method 
of  finding   the   function   which    corresponds   to   a   given 
differential. 

The  rules  for  the  differentiation  of  functions  are  explicit 
and  direct.  Those  for  determining  the  integral,  or  func- 
tion, from  the  differential  expression,  are  less  direct  and 
are  deduced  by  reversing  the  process  by  which  we  pass 
from  the  function  to  the  differential. 

208.  Let  it  be  required,  as  a  first  example,  to  integrate 
*.he  expression. 

xmdx. 

We  have  found  (Art.  32),  that 


whence, 


190  ELEMENTS    OF    THE 

xm+l 
and  consequently  --  , 

•  *  the  function  of  which  the  differential  is  xmdx. 

The  integration  is  indicated  by  placing  the  character  / 
Before  the  differential  which  is  to  be  integrated.  Thus, 
KG  write 


from  which  we  deduce  the  following  rule. 

To  integrate  a  monomial  of  the  form  xmdx,  augment 
the  exponent  of  the  variable  by  unity,  and  divide  by  the 
exponent  so  increased  and  by  the  differential  of  the 
variable. 

209.  The  characteristic  /  signifies  integral  or  sum. 
The  word  sum,  was  employed  by  those  who  first  used  the 
differential  and  integral  calculus,  and  who  regarded  the 
integral  of 

xndx 

as  the  sum  of  all  the  products  which  arise  by  multiplying 
the  mth  power    of    a?,  for  all  values  of  x,  by  the  con 
stant  dx. 

210.  Let  it  be  required  to  integrate  the  expression  —g. 
We  have,  from  the  last  rule, 

/(A/UU  /•   i  Q  X  X  JL 

_=/^.=___=_= 

In  a  similar  manner,  we  find 


INTEGRAL    CALCULUS.  191 

211.  It  has  been  shown  (Art.  22),  that  the  differential 
of  the  product  of  a  variable  multiplied  by  a  constant,  is 
equal  to  the  constant  multiplied  by  the  differential  of  the 
variable.     Hence,  we  may  conclude  that,  the  integral  of 
the  product  of  a  differential  by  a  constant,  is  equal  to  the 
constant  multiplied  by  the  integral  of  the  differential : 
that  is, 

/  axmdx  =  a  f  xmdx  =  a 

J  J 

Hence,  if  the  expression  to  be  integrated  have  one  or 
more  constant  factors,  they  may  be  placed  as  factors  with- 
out the  sign  of  the  integral. 

212.  It  has  also  been  shown  (Art.  22),  lhat  every  con- 
stant quantity  connected  with   the  variable  by  the   sign 
t)lus  or  minus,  will  disappear  in  the  differentiation ;  and 
hence,  the  differential  of  a  +  xm,  is  the  same  as  that  of 
x™ ;  viz.  mxm~ldx.     Consequently,  the  same  differential 
may  answer  to  several  integral  functions  differing  from 
each  other  in  the  value  of  the  constant  term. 

In  passing,  therefore,  from  the  differential  to  the  integral 
or  function,  we  must  annex  to  the  first  integral  obtained, 
a  constant  term,  and  then  find  such  a  value  for  this  term 
as  will  characterize  the  particular  integral  sought. 

For  example  (Art.  94), 

~  =  a,       or      dy  =  adxy 

CLCC 

is  the  differential  equation  of  every  straight  line  which 
makes  with  the  axis  of  abscissas  an  angle  whose  tangent 

is  a.     Integrating  this  expression,  we  have 

13 


192  ELEMENTS    OF    THE 

=  afdx, 


or  y  —  aX)  . 

or  finally,  y  =  ax  +  C. 

If  now,  the  required  line  is  to  pass  through  the  origin 
of  co-ordinates,  we  shall  have,  for 

x  =  0,       y  =  0,       and  consequently,      C  =  0. 

But  if  it  be  required  that  the  line  shall  intersect  the  axis 
of  Y  at  a  distance  from  the  origin  equal  to  +  6,  we  shall 
have,  for 

x  —  0,       y  =  -f  6,     and  consequently,     C  =  +  b  ; 
and  the  true  integral  will  be 

y  =  av-\-b. 

If,  on  the  contrary,  it  were  required  that  the  right  line 
should  intersect  the  axis  of  ordinates  below  the  origin,  we 
should  have,  for 

x  =  0,     y  s=a  —  b,     and  consequently,     C  =  —  b  ; 
and  the  true  integral  would  be 

y  =  ax  —  b. 

213.  It  has  been  shown  (Art.  95),  that 
xdx  +  ydy  =  0 

is  the  differential  equation  of  the  circumference  of  a  circle 
By  taking  the  integral,  we  have 

§,      or 
or  finally,  :r*  +  ^  +  C  =  0. 


INTEGRAL    CALCULUS.  193 

If  it  be  required  that  this  integral  shall  represent  a  given 
circumference,  of  which  the  radius  is  R,  we  shall  have, 
by  making 

a?  =  0,         y*=-C  =  R2, 
and  hence,  C=—  R2; 

and  consequently  the  true  integral  is 

0 

R2  =  0,       or 

The  constant  C,  which  is  annexed  to  the  first  integral 
that  is  obtained,  is  called  an  arbitrary  constant,  because 
such  a  value  is  to  be  attributed  to  it  as  will  cause  the 
required  integral  to  fulfil  given  conditions,  which  may  be 
imposed  on  it  at  pleasure. 

The  value  of  the  constant  must  be  such,  as  to  render 
the  equation  true  for  every  value  which  can  be  attributed 
to  the  variables. 

214.  There  is  one  case  to  which  the  formula  of  Art.  208 
does  not  apply.    It  is  that  in  which  m  =  —  1.    Under  this 
supposition, 

*W"M  --  ar*+1     -X°=  1  - 

+  l   ~     — 1  + 1~~    0  ~~    0   " 

But  when  m  =  —  1, 

dx 

fxmdx  =  fx~ldx  =  /  — , 
x 

and  C  —  =  log  a?  +  C.     (Art.  57). 

«/    x 

215.  Since  the  differential  of  a  function  composed  of 
several  terms,  is  equal  to  the  sum  or  difference  of  the  diffe- 
rentials (Art.  27),  it  follows  that  the  integral  of  a  differen- 


194  ELEMENTS    OF    THE 

tial  expression,  composed  of  several  terms,  is  equal  to  the 
sum  or  difference  of  the  integrals  taken  separately.  For 
example,  if 

du  =  adx  --  s-  -|-  x  -y/  x  dx,        we  have 
or 

fdu  =f(adx  --  3-  +  x  •v/tfcfo),         and 


216.  Every  polynomial  of  the  form 

(<z-f  bx  +  cx2^-  &c.)ndx, 


in  which  n  is  a  positive  and  whole  number,  may  be  inte- 
grated by  the  rule  for  monomials,  by  first  raising  the  poly- 
nomial to  the  power  indicated  by  the  exponent,  and  then 
multiplying  each  term  by  dx. 

If,  for  example,  we  make   n  =  2,   and  employ  but  two 
terms,  we  have 

f(a  +  bxfdx  =f(azdx  +  2a 


Integration  of  Particular  Binomials. 
217.  If  we  have  a  binomial  of  the  form 


that  is,  in  which  the  exponent  of  the  variable  without  iht 
parenthesis  is  less  by  unity  than  the  exponent  of  the  vari- 
able withint  we  may  make 


INTEGRAL    CALCULUS.  195 

a  -{-  bx"  =  zt     which  gives 

nbxn~ldx  =  dz,         or        xn~ldx  —  — ; 

no 

whence        du  =  zm-^-,         or         u  = 
nb 

and  consequently 


c 

(m+l)nb 

Hence,  the  integral  of  the  above  form,  is  equal  to  the  bino- 
mial factor  with  its  exponent  augmented  by  unity,  divided, 
by  the  exponent  so  increased,  into  the  exponent  of  the  vari- 
able within  the  parenthesis  into  the  coefficient  of  the 
variable. 

For  example, 

C;        and 


/(a  +  bx*Y  mxdx  =      (a  +  by?Y  +  C. 

oO 

218.  A  transformation  similar  to  that  of  the  last  article 
will  enable  us  to  integrate  certain  differentials  correspond- 
ing to  logarithmic  functions.  If  we  have  an  expression  of 
the  form 

adx 


du  = 


bx 


make     c  +  bx  =  z,     which  gives     dx  =  —  ,     and  by  sub- 
stituting, we  have 

adx          Cadz     a  Cdz 


196  ELEMENTS    OF    THE 

and  by  substituting  for  z  its  value 


In  a  similar  manner,  we  should  find 

/adx  a 

^te=-T 


in  which  the  integral  is  negative,  since  d(—x)=—dx. 

We  can  find,  in  a  similar  manner,  the  integral  of  every 
fraction  of  which  the  numerator  is  equal  to  the  differential 
of  the  denominator,  or  equal  to  that  differential  multiplied 
by  a  constant. 

If,  for  example,  we  have 

j    _(b  -\-2cx)  mdx  m 
a  +  bx  +  co?  ' 

make  tz  +  bx  -{-  ex2  =  z,    which  gives,    bdx  -f  Zcxdx  =  dz, 
and  hence, 

mdz 
du  =  -  ,     or    u  =  m\ogz, 

and  by  substituting  for  z  its  value 

u  =  mlog(a  +  bx  -f  ex2). 

Of  Differentials  whose  Integrals  are  expressed  by 
the  Circular  Functions. 

219.  We  have  seen,  Art.  71,  that  if  x  designates  an  arc 
and  u  the  sine,  to  the  radius  unity,  we  shall  have 

du 


hence' 


INTEGRAL    CALCULUS.  197 

du 


or  adopting  the  notation  of  Art.  72, 
du 

V^tf 

If  the  arc  expressed  in  the  second  member  of  the  equa- 
tion be  estimated  from  the  beginning  of  the  first  quadrant, 
the  sine  will  be  0,  when  the  arc  is  0,  and  we  shall  have, 
for  u  =  tf 

—==.  —  0,     and  consequently     C  =  0, 
VI  —u2 

and  under  this  supposition,,  the  entire  integral  is 

du            .     i 
~lu. 


To  give  an  example,  showing  the  use  of  the  arbitrary 
constant,  let  us  suppose  that  the  arc  which  is  to  be  ex- 
pressed by  the  second  member  of  the  equation,  is  to  be 
estimated  from  the  beginning  of  the  second  quadrant.  This 
supposition  will  render 

du 


But  when    u  =  l9     sm~lu  =  — v  ;     hence, 

— *-+C  =  0,       or       C=—  —  *•: 
2  2 

and  we  have,  for  the  entire  integral,  under  this  supposition, 
du  I 

— :  =  Sin      U —if. 


198 


ELEMENTS    OF    THE 


220.  It  frequently  happens  that  we  have  expressions  to 
integrate  of  the  form 

dzt 

Let  us  suppose,  for  a  moment,  that  a  is  the  radius  of  a 
circle,  and  z  the  sine  of  any  arc  of  the  circle ;  and  that  u 
is  the  sine  of  an  arc  containing  an  equal  number  of  degrees 
in  a  circle  whose  radius  is  unity :  we  shall  then  have, 

1    :   u   :  :   a   :  z : 


'     \ 
hence,                 u  = 

and  consequently, 
r     du 

—  ,       and       di 
a 

-       dz    -^ 

c    a 

dz 
'  =  T; 

r        dz 

J  VT=tf- 

f\m\ 

J  Va*-z*' 

hence, 


du 


dz 


/du  f      az  .     l  z 

,  —    I  =  sin"1  — 

Vl-u2     J  Va2-z2  a 


the  arc  being  still  taken  in  a  circle  whose  radius  is  unity. 

221.  We  have  seen  (Art.  71),  that  if  x  designates  an 
arc,  and  u  the  cosine,  to  the  radius  unity,  we  shall  have 

du 


hence, 


I- 


du 


l-u2 

or  adopting  the  notation  of  Art.  72, 
du 


INTEGRAL    CALCULUS.  199 

If  the  arc  be  estimated  from  the  beginning  of  the  first 

quadrant,  it  will  be  equal  to  — if  for  u  —  0 ;  hence,  the 

2  I 

first  member  of  the  equation  becomes  equal  to  — v  when 

1 

u  =  0.     But  under  this  supposition, cos"1  u=  — *  :    hence, 

C  =.  0,  and  the  entire  integral  is 


222.  By  a  method  analogous  to  that  of  Art.  220,  we 
should  find 


the  arc  being  estimated  to  the  radius  unity. 

223.  We  have  seen  (Art.  71),  that  if  x  represents  an 
arc,  and  u  its  tangent,  to  the  radius  unity,  we  have 

du 


.    du 
hence, 


or,  adopting  the  notation  of  Art.  72, 


If  the  arc  is  estimated  from  the  beginning  of  the  first 
quadrant,  we  shall  have 


when 


•ffar*t      hence,      C  =  0, 


200  ELEMENTS    OF    THE 

and  the  entire  integral  is 

/du 
TT^  =  tang""- 

224.  To  integrate  expressions  of  the  form 
dz 


let  us  suppose  for  a  moment  that  a  is  the  radius  of  a  circle, 
and  z  the  tangent  of  any  arc,  and  that  u  is  the  tangent 
of  an  arc  containing  an  equal  number  of  degrees  in  a  circle 
whose  radius  is  unity  :  we  shall  then  have,  as  in 

(Art.  220), 

1    :   u   :  :   a  :   z  ; 

.  z  2     zz  j       dz 

hence,  u  =  —  ,       u2  =  —  ,       and      du  =  —  , 

a  a  a 

and  consequently, 

du  r    dz  _j  z 


hence,  by  dividing  by  a, 
dz 


the  arc  being  estimated  to  the  radius  unity. 

225.  We  have  seen  (Art.  71),  that  if  x  represents  an 
arc,  and  u  the  versed-sine,  to  the  radius  of  unity,  we  have 

du 
dx  =     ,  2; 

hence,          /     ,  =.  —  oc  =  ver-sin~  lu  -}-  C : 

J  V2u  —  u2 


INTEGRAL    CALCULUS.  201 

and  if  the  arc  is  estimated  from  the  beginning  of  the  first 
quadrant,  C  =  0,  and  we  shall  have 


=  ver-sm~  u. 


226.  To  integrate  an  expression  of  the  form 

dz 

V2az-z2' 

Suppose,  as  before,  a  to  be  the  radius  of  a  circle,  and 
we  shall  have  (Art.  220), 

z  ,        dz 

u  =  —t        du  =  — ; 
a  a 


du  r        dz  ,  z 

=  ver-sin-1  —  . 


and  consequently, 

/du  r 

.  =  I     . 

V2u  —  u2      J  V2az  — 

to  the  radius  unity. 

Integration  by  Series. 

227.  Every  expression  of  the  form 
Xdx, 

in  which  X  is  such  a  function  of  #,  that  it  can  be  developed 
in  the  powers  of  #,  may  be  integrated  by  series. 
.  For,  let  us  suppose 

X  =  Ax"  +  Bxb  +  Caf  +  Dxd  +  &c.,     then, 
Xdx  =  Aafdx  +  Bafdx  4-  Cxedx  +Dx*dx  +  &c., 


202  ELEMENTS    OF    THE 

Hence,  the  integration  by  series  is  effected  by  develop- 
ing the  function  X  in  the  powers  of  x,  multiplying  the 
series  by  dx,  and  then  integrating  the  terms  separately. 

Let  us  take,  as  a  first  example,     -  , 

a  +  x 

-  =  dx  x  -  =  dx(a  +  x)"\ 
a  +  o?  a  +  07 

/  \il         x        a?       sP  .    » 

(a  +  a;)-.  =  ___  +  ___  +  &c.; 

and  consequently, 

/dx         Cf  1    ,        xdx  ,  x2dx      x3dx  ,    - 
_  =      (  —  dx-—r  +  —  ---  r  +  & 
a  +  x     J  \  a  a2          a3  a4 

and  integrating  each  term  separately,  we  obtain 

dx         x        x2         x         a*        &  c 


and  remarking  that      /  —  -    —  =  log  (a  +  a?)       (Art.  218), 
«/    d  ~\  oc 

we  have 


To  determine  the  value  of  the  constant,  make   a?=0, 
which  gives 

log  a  =  0  +  C,       or       C  —  log  a  ;     hence, 

*          i  CC  CCt  OC  CO  p 

a;)  =  loga  +  --—  2  +  ^-—  4+&c., 


log(a+«)  -  log«=log    l  +        =  ..  -        +        -  &c., 
a  result  which  agrees  with  the  development  in  Art.  58. 


INTEGRAL    CALCULUS.  203 

dx 

228.  Let  us  take,  for  a  second  example     -  -. 

1  +  of 

flr 
We  have, 


and  by  developing  and  integrating, 


When  we  make  x  =  0,  the  arc  is  0  ;  hence, 

x*      a?      x7 
tang  lx=x  —  —  -  +  •—  —  —  -f  &c.; 

oO/ 

a  result  which  corresponds  with  that  of  Art.  78. 

229.  If,  in  the  expression    —  —  ,    we  place  a?  in  the 

1  -f-  3T 

first  term  of  the  binomial,  and  then  develop  the  binomial 
,  we  obtain 


C    dx          r/l        1        1        1 

I  -2  —  -=  /  (-3---T  +  -e--» 

J  a^  +  1       J  \a?      #4      a?6       a?8 

and  by  integrating,  we  have 


To  find  the  value  of  the  constant  C,  let  us  make  the 
arc  =  90°  =  «:*••     This  supposition  will  render  the  tan- 

gent x  infinite,  and  consequently  every  term  of  the  series 
will  become  0,  and  the  equation  will  give 


--T  =  o  +  c,    or 

2 


204  ELEMENTS    OF    THE 

Making  this  substitution,  we  have,  for  the  true  integral, 


dx    _tano.-i-rr=JL^_J_4-J_       l        fcc 


230.  The  two  series,  found  from  the  expressions  —  -  —  - 

dx  l  +  x* 

and    -=  -  ,    are,  as  they  should  be,  essentially  the  same. 

ar  +  1 
For,  the  tangent  of  an  arc  multiplied  by  its  cotangent, 

is  equal  to  radius  square   or  unity  (Trig.  Art.  XVIII). 

Hence,  if  we  substitute  for  #,  in  the  first  series,   -,  we 

x 

shall  have,  for  the  complemental  arc, 

_i  1        1          1  1 


and  subtracting  both  members  from    -«r, 


=  tang-o;  =  -       -      +       _  _  _+  &c. 
231.  We  have  found  (Art.  71), 


VI  — 

and  by  developing,  we  find 


multiplying  by  dx,  and  integrating,  we  obtain, 

1  or3        1     3  x5        1     3     5  a?7 
sm-^  =  ,  +  __  +  „.__.  +  _._._- 


INTEGRAL    CALCULUS.  205 

tti<5  constant  being  0  when  the  arc  is  estimated  from  the 
beginning  of  the  first  quadrant. 

If  we  take  the  arc  of  30°,  the  sine  of  which  is  equal 
to  half  the  radius  (Trig.  Art.  XIV),  we  shall  have 


1,1111311        13511 
_  +  i._._  +  _._.__  +  _._._._._ 


hence, 


and  by  taking  the  first  ten  terms  of  the  series,  we  find 

«•  =  3.  1415962, 

which  is  true  to  the  last  decimal  figure,  which  should  be  5. 
232.  We  will  add  a  few  more  examples. 

1.  To  integrate  the  expression  -  . 

V  x  —  a? 

By  making     V~x  =  u,     we  have 

dx       _          (lx          _      2du 
Vx-a?~  V~xVl-x~  V\-u2' 

But  from  the  last  series 

2du  r      In?    ,  1  Su5   ,  1  3  5w7  , 

h-.  ---  !--•-•  ---  h&c. 
23  ^2  45  ^2  4  67^ 


2. 


206  ELEMENTS    OF   THE 

But 


hence 


LJL  11  L    1-1-8  *    &c 

22a       2  '  4  4o2       2.4.6  Sa3 


5_ 

1     2  a?8        112 


and  consequently 

xi         1     1    a?         1     1     1    a? 

_!  J_  JL    l    *3 
~  2  '  4  '  6  '  9  8a3~ 

If  the  radius  of  a  circle  be  represented  by  a,  and  the 
origin  of  co-ordinates  be  placed  in  the  circumference,  the 
equation  will  be  (An.  Geom.  Bk.  Ill,  Prop.  I,  Sch.  3), 


y2  —  2ax  —  a?;     hence    y  = 
and  consequently  (Art.  130) 


dx  v  2  ax  —  o?  —  ydx 

is  the  differential  of  a  circular  segment. 

If  we  estimate  the  area  from  the  origin,  where  x  —  0, 
we  shall  have  0  =  0.  If  then  we  make  a?  =  a,  the  series 
will  give  the  area  of  one  quarter  of  the  circle,  if  we  maite 
a?  —  2 a,  of  the  semicircle. 

1.3.5a?7   . 


INTEGRAL    CALCULUS.  207 

dx  1  1.3  1.3.5 

~  *~~ 


Integration  of  Differential  Binomials. 

234.  Differential  binomials  may  be  represented  under 
the  general  form 


in  which,  without  affecting  the  generality  of  the  expres- 
sion, m  and  n  may  be  regarded  as  entire  numbers,  and  n 
as  positive. 

For,  if  m  and  n  were  fractional,  and  the  binomial  of 
the  form 


make  x  =  z6,  that  is,  make  the  exponent  of  z  the  least 
common  multiple  of  the  denominators  of  the  exponents 
of  x,  and  we  shall  then  have 


. 
x3  dx(a  +  bx*)*  =  Qz7dz(a 

in  which  the  exponents  of  the  variable  are  entire 
If  n  were  negative,  we  should  have, 


and  by  making    x  =  —  ?     we  should  obtain 


—  s— 

the  same  form  as  before. 


208  ELEMENTS    OF    THE 

Furthermore,  the  binomial 

p_ 
xm~ldx(axr-{-bxn)<' 

may  be  reduced  to  the  form 


by  dividing  the  binomial  within  the  parenthesis  by  stf,  and 

PT 

multiplying  the  factor  without  by    x  q  . 

235.    Let  us  now  determine   the   cases  in  which  the 

JO 

binomial     xm~ldx(a-\-  bxn)q     has  an  exact  integral. 
Make     a  -f  bxn  —  zq\     we  shall  then  have 


=          -, 

and  by  differentiating, 


-- 

nb 
hence 


- 
no 


which  will  have  an  exact  integral  in  algebraic  terms  when 

—  is  a  whole  number  and  positive  (Art.  216).     If   —  is 
n  n 

negative  see  Art.  260. 

Hence,  every  differential  binomial  has  an  exact  inte- 
gral, when  the  exponent  of  the  variable  without  the  paren 
thesis   augmented   by  unity,    is   exactly  divisible   by  the 
exponent  of  the  variable  within. 

Thus,  for  example,  the  expression 


INTEGRAL    CALCULUS.  209 

has  an  exact  integral.     For,  by  comparing  it  with  the 
general  binomial,  we  find 

m  =  6,  n  —  2,      and  consequently,     —  =  3, 
and  the  transformed  binomial  becomes 


236.  There  is  yet  another  case  in  which  the  binomial 

p_ 
xm~ldx(a-{-bxn)i  has  an  exact  integral. 

If  we  multiply  and  divide  the  quantity  within  the  paren 
thesis  by  xn,  we  have 


xm'ldx(a  +  Wp  =  x>n~l  dx[(ax~n 

L    IS 
=  xm~ldx(ax~n 


Now,  if  we  add  unity  to  the  exponent  of  x  without  the 
parenthesis,  and   divide  by     —n,     the  quotient  will  be 

—  (  ---  h  —  X     and   the   expression  will   have    an   exact 

integral  when  this  quotient  is  a  whole  number  (Art.  235). 

Hence,  every  differential  binomial  has  an  exact  integral, 
when  the  exponent  of  the  variable  without  the  parenthesis 
augmented  by  unity  and  divided  by  the  exponent  of  the 
variable  within  the  parenthesis,  ^plus  the  exponent  of  the 
parenthesis,  is  an  entire  number. 

237.  The  integration  of  differential  binomials  is  effected 
by  resolving  them  into  two  parts,  of  which  one  at  least  has 
a  known  integral. 

We  have  seen  (Art.  28)  that 

d(uv}  =  udv  -f  vdu, 


210  ELEMENTS    OF    THE 

whence,  by  integrating, 

uv  =fudv  -}-fvdu, 
and,  consequently, 

fudv  =  uv  —fvdu. 

Hence,  if  we  have  a  differential  df  the  form  Xdx,  in 
which  the  function  X  may  be  decomposed  into  two  factors 
P  and  Q,  of  which  one  of  them,  Qdx,  can  be  integrated, 
we  shall  have,  by  making  /  Qdx  =  v  and  P  =  ut 

fPQdx  =  Pv-fvdP, 

in  which  it  is  only  required  to  integrate  the  term  fvdP. 
238.  To  abridge  the  results,  let  us  write  p  for  —  ,  in 

which  case  p  will  represent  a  fraction,  and  the  differential 
binomial  will  take  the  form 


If  now,  we  multiply  by  the  two  factors  xn  and  a?~n,  the 
value  will  not  be  affected,  and  we  obtain 


Now,  the  factor  xn~ldx(a  +  bxn)p  is  integrable,  whatever 
be  the  value  of  p  (Art.  217)  ;  and  representing  this  factor 
by  dv,  we  have 


and,  consequently, 


ni-n 


INTEGRAL    CALCULUS.  211 

But,  fxm~n-ldx(a  +  bxn)p+l  = 

fxm-n~ldx(a  +  bxn)p(a  +  bxn)  = 
afxm-n-ldx(a  +  bxnY  +  bfxm~ldx(a  +  bxn)p', 


substituting  this  last  value  in  the  preceding  equation,  and 
collecting  the  terms  containing  the  integral 

fxm-ldx(a+bxn)», 
we  have 


xm~n(a  +  bxn)p+l  -  a(m  -  n)fxm-n~ldx(a  +  bx)p  . 
(p+l)nb 

whence, 

formula  (A.)  ..........  fxm~ldx(a  +  bxn)p  = 

xm-n(a  +  bo?)'*  l  -g(m-  n)fxm~n-ldx(a  +  bxn}*> 
b(pn-\-  m) 

This  formula  reduces  the  differential  binomial 


fxm-ldx(a  +  bxn)p     to  that  of    fxm-n~ldx(a  +  bxn)f  ; 
and  by  a  similar  process  we  should  find 
fxm-n-ldx(a  +  bxn)p  to  depend  on  fxm~'2n-ldx(a-\-bxny>; 

and  consequently,  each  process  diminishes  the  exponent 
of  the  variable  without  the  parenthesis  by  the  exponent 
of  the  variable  within. 

After  the  second  integration,  the  factor  m  —  n,  of  the 
second  term,  will  become  m  —  2n  ;  and  after  the  third, 
m  —  3n,  &c.  If  m  is  a  multiple  of  TI,  the  factor  m  —  n, 
m  —  2n,  m  —  3?z,  &c.,  will  finally  become  equal  to  0,  and 
then  the  differential  into  which  it  is  multiplied  will  disap- 


212  ELEMENTS    OF    THE 

pear,  and  the  given  differential  will  have  an  exact  integral 
which  corresponds  with  the  result  of  Art.  235. 

239.  Let  us  now  determine  a  formula  for  diminishing 
the  exponent  of  the  parenthesis. 
We  have 

fxm~ldx(a  +  bxnY  =  fxm-ldx(a  +  bxn)p~l(a  +  bxn)  = 
afxm~ldx(a 


Applying   formula  (A)  to   the   second   term,  by  placing 
m  4-  n  for  m,  and  p  —  1  for  p,  we  have 


xm(a  4-  bxn)p  -  amfxm-ldx(a  +  bxn)p~l 
b(pn  4-  m) 

Substituting  this  value  in  the  last  equation,  we  have 

formula  (B)  ................  fxm~ldx(a  +  bxn)f  = 

xm(a  +  bxnY  +  pnafxm~ldx(a  +  bxn)p~l 
pn  +  m 

which  diminishes  the  exponent  of  the  parenthesis  by  unity 
for  each  integration. 

240.  By  means  of  formulas  (A)  and  (B),  we  reduce 

to 


rn  being  the  greatest  multiple  of  n  which  can  be  taken 
from  m—  1,  and  s  the  greatest  whole  number  which  can 

be  subtracted  from  p. 

^ 

For  example,    fx7dx(a  +  &r3)2     is  reduced,  by  formula 

(A),  to 

i. 
fx*dx(a  +  bx*Y>     and  then  to     f  xdx(a 


INTEGRAL    CALCULUS.  213 

£ 

and  by  formula  (B)    Jxdx(a  -f  for3)2,     reduces  to 

1  1 

fasdx  (a  +  bx*)  2  ,     and  finally  to    fxdx  (a  +  bx3)  2  . 

241.  It  is  evident  that  formulas  (A)  and  (B)  will  only 
diminish  the  exponents  m  —  1  and  p,  when  m  and  p  are 
positive.  We  will  now  determine  two  formulas  for  dimin- 
ishing these  exponents  when  they  are  negative. 

We  find  from  formula  (A) 


xm~n(a  +  bxn)p+l  -  b(m  +  np)fvm-ldx(a 
a(jri  —  ri) 

and  placing  for  m,   —  m  +  n,  we  have 

formula  (C)  .............  fx~m-ldx( 

x~m(a  +  bx*)p+l  +  b(m  —  n  —  np\f  x~m+n-ldx(a  +  bo?)* 


—  am 


in  which  formula,  it  should  be  remembered  that  the  nega- 
tive sign  has  been  attributed  to  the  exponent  m. 

242.  To  find  the  formula  for  diminishing  the  exponent 
of  the  parenthesis  when  it  is  negative. 

We  find,  from  formula  (B), 


xm(a  +  bxn)p  —  (m-\-  np)fxm~ldx(a  +  la*)* 
pna 

writing  for  p,  —  p  +  1,  we  have 

formula  (D)  .............  fxm-1dx(a  +  bxn)~p  = 

xm(a  +  bxn)~p+l  —  (m  +  n  —  np)fxm~ldx(a  +  bxn)~p+l 
(p—  l)na 


214  ELEMENTS    OF    THE 

This  formula  does  not  apply  to  the  case  in  which  p  =  l. 
Under  this  supposition,  the  second  member  becomes  infi- 
nite, and  the  differential  becomes  that  of  a  transcendental 
function. 

243.  It  is  sometimes  convenient  to  leave  the  variable  in 
ooth  terms  of  the  binomial.  We  shall  therefore  determine 
a  particular  formula  for  integrating  the  binomial 


This  binomial  may  be  placed  under  the  form 

—  JL  _L 

fa?   ~*dx(2a-x)~t, 

and  if  we  apply  formula  (A),  after  making 


—  ,    n=l,    p  —  —  —  ,    a  =  2a,    b  =  -  1, 


we  shall  have 


9  ? 

and  if  we  observe  that 

9-1       7-1   1       ?--       9-1   -1 
x     2  =x     x2      x     *=x     x    % 

and  pass  the  fractional  powers  of  x  within  the  parentheses 
we  shall  have 

formula  (E) 

/2ax-x*  t  (2q-l)a    f 

V2ax-x?' 


INTEGRAL    CALCULUS.  215 

which  diminishes  the  exponent  of  the  variable  without  the 
parenthesis  by  unity.  If  q  is  a  positive  and  entire  num- 
ber, we  shall  have,  after  q  reductions 


Integration  of  Rational  Fractions. 

244.  Every  rational  fraction  may  be  written  under  the 
form 

Pa?-'+Qa?-g....  +RX  +  S. 
P'xn     +QV-1  ____  +R!x+S'  ' 

in  which  the  exponent  of  the  highest  power  of  the  varia- 
ble in  the  numerator,  is  less  by  unity  than  in  the  denomi- 
nator. For,  if  the  greatest  exponent  in  the  numerator  was 
equal  to  or  exceeded  the  greatest  exponent  in  the  denomi- 
nator, the  division  might  be  made,  giving  one  or  more 
entire  terms  for  a  quotient  and  a  remainder,  in  which  the 
exponent  of  the  leading  letter  would  be  less  by  at  least 
unity,  than  the  exponent  of  the  leading  letter  in  the  divisor. 
The  entire  terms  could  then  be  integrated,  and  there 
would  remain  the  fraction  under  the  above  form. 

Place  the  denominator  of  the  fraction  equal  to  0  :  that 
is,  make 

PV+  QV-1  ......  R'x+  S'  =  0, 


and  let  us  also  suppose  that  we  have  found  the  n  binomial 
factors  into  which  it  may  be  resolved  (Alg.  Art.  264). 
These  factors  will  be  of  the  form  x  —  a,  x  —  b,  x  —  c, 
x  -d,  &c.  Now  there  are  three  cases  : 


216  ELEMENTS    OF    THE 

1st.   When   the   roots   of   the    equation   are   real   and 
unequal. 

2d.  When  they  are  real  and  equal. 
3d.  When  there  are  imaginary  factors. 
We  will  consider  these  cases  in  succession. 

1st.   When  the  roots  are  real  and  unequal. 

245.  Let  us  take,  as  a  first  example,     — -. 

By  decomposing  the  denominator  into  its  factors,  we 
have 

adx     _  adx 

a?  —  a2  ~  (x  —  a)  (x  -f  a) ' 

and  we  may  make 

adx  /A  B    \  , 

L  I    _^I_  ^^^___    1  rj  /y» 

(x  —  a)  (x  +  a)       \x  —  a      x-{-aJ 

in  which  A  and  B  a.re  constants,  whose  values  are  yet  to 
be  determined.  In  order  to  determine  these  constants, 
let  us  reduce  the  terms  of 'the  second  member  of  the 
equation  to  a  common  denominator ;  we  shall  then  have 

adx  (Ax  +  Aa  -f-  Bx  —  Ba)  dx 


(x  —  a)(x  +  a)  (x  —  a)( 

In  comparing  the  two  members  of  the  equation,  we  find 

a  =  Ax  +  Aa  +  Bx  —  Ba ; 
or  by  arranging  with  reference  to  a?, 

(A  +  B)x  +  (A-B-l)a  =  0. 
But,  since  this  equation  is  true  for  all  values  of  x}  the 


INTEGRAL    CALCULUS.  217 

coefficients  must  be  separately  equal  to  0  (Alg.  Art.  208)  : 
hence 

A  +  B  =  0,     and    (A-B-l)a  =  0, 
which  gives 


Substituting  these  values  for  A  and  B,  we  obtain 

adx         -ndx        fdx 
a?  —  a2     x  —  a      a?  -f  a  ' 

and  integrating,  we  find  (Art.  218) 

/cidx          1  1 

rf—tf  =  Ylos(*  ~  a)  -  ylos(*  +  fl)  +  c> 

and,  consequently, 

/adx         1  .     fx  —  a 
?^==¥1°sfe 


246.  Let  us  take,  as  a  second  example,     ^-— dx. 

(TX  —  3? 

The  factors  of  the  denominator  are  x  and  a2  —  x2 ;  but 

hence,  the  given  fraction  becomes 

dx. 


x(a-x)(a-}-x') 
Let  us  now  make 


=^+-*-+  ° 


x(a  —  x}(a  +  x)        x       a  —  x      a 


218  ELEMENTS    OP    THE 

reducing  the  terms  of  the  second  member  to  a  common 
denominator,  we  have 

a?  +  bx2  Act2-  Aoc?+  Bax  +  Bx2+  Cax  -Cx2 


x(a  —  x}(a-i-x)  x(a  —  x)(a 

and,  comparing  the  like  powers  of  x  (Alg.  Art.  208), 


From  these  equations,  we  find 


=  a,        =  ,        =--. 

and  substituting  these  values,  we  obtain 

a3  +  bx2  i          dx         a  -4-b     7  a  4-b      , 

—  -  -dx  —  a  --  1  --  -  —  I  -  -dx  --    -  -  -dx  : 
a2x  —  x*  x       2a  —  cc  2a      x 


and  integrating  (Art.  218), 


-  -  [log(a  —  a?)  +  log(a  +  0:)]  +  C 
=  alog,r  ---  l°g(a  —  #)  (a  +  x)  +  C 


=  aloga?  —  (a  +  6)  log  -y/a2  —  x2  +  C. 

QX  _  ^ 

247.  Let  us  take,  for  a  third  example,     —  --  dx. 

cc?  _  6  x  -f  8 


INTEGRAL    CALCULUS.  219 

Resolving  the  denominator  into  the  two  binomial  factors 
(Alg.  Art.  142),  (x  —  2),  (a?  — 4),  we  have 

3  a;  —  5  A  B 

—- — — —  = hence 

x2  —  6x  +  8       x  —  2       x  —  ± 

3x  —  5          Ax  —  4A+Bx  —  2B 


"~  a?  —  607  +  8 

and  by  comparing  the  coefficients  of  x,  we  have 
-5=-4A-2£,         3  =  A  +  B, 
which  gives 

,.2,  ,__£.,,_ 

• 

and  substituting  these  values,  we  have 


r    3 

J  a?- 


337-5     ,        j_  r  dx      7  r  dx 

~}~~ 


248.  Let  us  take,  as  a  last  example, 

xdx 
x2  -\-kax  —  W 

Resolving  the  equation 


we  find 

x—  —  2a  +  V4a2  +  tf,         x=—2a  — 
and  consequently,  for  the  product  of  the  factors, 
r)(a?+2a- 


220  ELEMENTS    OF    THE 

To  simplify  the  work,  represent  the  roots  by  —  K  and 
—  Ly  and  the  factors  will  then  be 


K,    x 

and  we  shall  have 


x  A  B 


^2  —  — r"^  H rr- :    hence 

—  b2      x  +  K      x  +  L 

x  Ax  +  AL  +  5o?  f  £# 


_  _  _ 

x2  -f  4  ax  —  b2  "  x2  +  4  ax 

whence, 


and,  consequently, 

K  E_         L 

~' 


K-L'  K-L' 

hence, 

.-  log(X+L)+C. 
—  bz     K—L  K  —  L 

249.  In  general,  to  integrate  a  rational  fraction  of  the 
form 

-'  ____  +Rx+S  d 


1st.  Resolve  the  fraction  into  m  partial  fractions,  of 
which  the  numerators  shall  be  constants,  and  the  denomi- 
nators factors  of  the  denominator  of  the  given  fraction. 

2d.  Find  the  values  of  the  numerators  of  the  partial 
fractions,  and  multiply  each  by  dx. 


INTEGRAL    CALGLLUS.  221 

3d.  Integrate  each  partial  fraction  separately,  and  the 
sum  of  the  integrals   thus  found  will   lie  the   integral 


sought. 


250.  The  method  which  has  just  been  explained,  will 
require  some  modification  when  any  of  the  roots  of  the 
denominator  are  equal  to  each  other.  When  the  roots  are 
unequal,  the  fraction  may  be  placed  under  the  form 


(x  —  a)(x  —  b)  (x  —  c)(x—  d)  (x  —  e) 

A  +  * +-C-+-A+  E 


x  —  a      x  —  b      x  —  c     x  —  d     x  —  ey 

if    several   of    these   roots    are   equal,   as    for   example, 
a  =  b  =  c,  the  last  equation  will  become 


Px*  +  Q*3  4-  &c.          A  +  B  +  C         D  E 

H 


(x  —  a)3  (x  —  d)  (x  —  e)  x  —  a  x  —  d      x  —  e' 

in  which  A  +  B  +  C  may  be  represented  by  a  single  con- 
stant A! .  t 

Now,  in  reducing  the  second  member  of  the  equation  to 
a  common  denominator  with  the  first,  and  comparing  the 
coefficients  of  the  like  powers  of  a?,  we  shall  have  five 
equations  of  condition  between  three  arbitrary  constants, 
A',  D,  and  E :  hence,  these  equations  will  be  incompati- 
ble with  each  other  (Alg.  Art.  103). 

If,  however,  instead  of  adding  the  three  partial  fractions 

ABC 


a?        x  —  a'        x  —  d' 


which  have  the   same  denominator,  we  go  through  the 


222  ELEMENTS    OF    THE 

process  of  reducing  them  to  one,  their  sum  may  be  placed 
under  the  form 

A'  +  B'x+C'a* 
(x-of        > 

or,  by  omitting  the  accents, 

A  +  Bx  +  Cx2 
(x-a)* 

251.  Let  us  now  make 

x — a  =  z,     and  consequently,     x  =  z-\-a] 
we  shall  then  have 

A  +  Bx  +  Cx2  _  A  +  Ba  +  Co2  +  Bz  +  2Caz  +  C# 
(x-a)3  ~^r~ 


A  +  Ba+Ca2      B -\-2Ca  ,    C 
—  — ! _ -i _ ; 

z*  z2  z 

substituting  for  z  its  value,  and  representing  the  numera- 
tors by  single  constants,  we  have 

A  +  Bx+Ca?  =      A1  B'       |      Cr 

the  form  under  which  the  fraction  may  be  written. 

Since  the  same  reasoning  will  apply  to  the  case  iu 
which  there  are  m  equal  factors,  we  conclude  that 

PaT-1 4-  Qxm~z  .       .  +  Rx  4-  S 


A  A!  A!'  A!r.  .  ! 

x-am-x-~ x-a 


252.  In  order,  therefore,  to  integrate  the  fraction 

p^*_|_  QV+  &c.      , 
(^-a)3"^^^)^-^  *' 


INTEGRAL    CALCULUS.  223 

place  it  equal  to 


(x-a)3  ^(x-a)2^  x-a^  x-d  l  x-e' 

then,  reducing  to  a  common  denominator,  and  comparing 
the  coefficients  of  the  like  powers  of  x,  we  find  the  values 
of  the  numerators  of  the  partial  fractions.  Multiplying 
each  by  dx,  and  the  given  fraction  may  be  written  under 
the  form 


. 

(x—a)2          (x—a)          x—d          x  —  e 

The  first  two  fractions  may  be  integrated  by  the  method 
of  Art.  217,  and  the  three  last  by  logarithms.  Hence,  finally, 


r 
J 


rf 


(x-a)3(x-d)(x-e)  2(x-a)2     x-a 

+  A"log(ar  —  a)  +  jDlog(a?  —  d)  +  E\og(x  —  e)  -f  C. 

253.  Let  it  be  required  to  integrate  the  fraction 
2  ax  '   , 


We  have 

2  ax 


(x  4-  a)       (a7-far     #-fa 

reducing  the  fractions  of  the  second  member  to  a  common 
denominator,  and  comparing  the  coefficients  of  x  in  the 
two  members,  we  have 

2a  =  Af     and     A-\-Afa  —  0: 

hence, 

A=  —  2a2,     and    A/  =  2a; 
15 


224  ELEMENTS    OP    THE 

and,  consequently, 

2axdx  2a2dx          2adx 

}2~  ~(#  +  a)2+  (x  +  a)' 


hence,  (Arts.  217  &  218), 

/2axdx          2  a? 
T=^ 


254.  Let  us  find  the  integral  of 
x2dx 


x3  —  ax2  —  a2x  -\-  a?  * 

By  placing  the  denominator  equal  to  0,  we  see  that,  by 
making  x  =  a,  the  terms  will  destroy  each  other  :  hence,  a 
is  a  root  of  the  equation,  and  x  —  a  a  factor.  Dividing  by 
x  —  a,  the  quotient  is  x2  —  a2:  hence,  the  fraction  may  be 
placed  under  the  form 


(a?-a2)(x-a)  ~  (x  +  a)(x-  a)(x-a) 


Let  us  now  make 

x2  A  A'  B 


(*-«) 

Reducing  the  terms  of  the  second  member  to  a  common 
denominator,  we  have 

x2  =  A(x-\-a)-\-Ar(x2-a2)-\-B(x-d)\ 

(x-a)2(x  +  a)~  (x-  a)2(x  +  a) 

and  developing,  and  comparing  the  coefficients  of  the  like 


INTEGRAL    CALCULUS.  ^  225 

powers  of  a?,  we  obtain  the  equations 

A'  +  B=1,         A-2Ba  =  0,         Aa  -  A'a2  +  Be?  =  0. 

Multiplying  the  first  equation  by  a2,  and  adding  it  to  the 
third,  we  have 


then  multiplying  the  second  by  a,  and  adding  it  to  the  last, 
we  have 

a2  =  2  Aa,     and  consequently,     A  = — a  ; 
substituting  this  value  of  A,  we  find 

T>  1  At  3 

B  =  —     and      A'  =  — . 
4  4 

Substituting  these  values  of  A,  A',  and  B,  we  have 
a<£r  3c?o?  <£r 

2/          i       -\  /-»/..          ~"-\5>.  "l"     ^   / 


(x-a)2(x 
and  consequently, 

/x*dx  a  3  ,     ,          . 

xt-axt-cfx  +  a^    -^T^(x 


255.  We  can  integrate,  in  a  similar  manner,  when  the 
denominator  contains  sets  of  equal  roots.  Let  us  take,  as 
an  example, 

adx  adx 


226  ELEMENTS    OF   THE 

Make 

a  A  Af  B  B1 


#-   1)2(^+1)2         (ay.  1)8         a;-!    ^  (tf  +  1  )*  ^  0?  +  1  ' 

reducing  the  second  member  to  a  common  denominator, 
we  find  the  numerator  equal  to 


and  comparing  the  coefficients  with  those  of  the  numera- 
tor of  the  first  member,  we  have  the  following  equations  : 

A'  +  B'  =  0, 
A  +  A'+    B-B'  =  0, 
%A  -Af-2B-Bf  =  0, 
A  -  A!  +   B  +  B'  =  a. 

Combining  the  first  and  third  equations,  we  find  A  =  B; 
and  combining  the  second  and  fourth,  gives  2  A  -f  2B  =  a: 
hence,  we  have 

A       T>      a         At          a         R/      a  . 
=  T'  "T  =TJ 

consequently,  thp  given  differential  becomes 

dx  dx  dx  dx 


and  by  integrating, 


256.  If  an  equation  of  the  second  degree  has  imaginary 
roots,  the  quantity  under  the  radical  sign  will  be  essentially 


INTEGRAL    CALCULUS.  227 

negative  (Alg.  Art.  144),  and  the  roots  will  be  of  the  form 
x  =  =F  a  +  b  V—  1,  x  =  +  «  —  b -y/  —  1, 

and  the  two  binomial  factors  corresponding  to  the  roots 
will  be 

( x  ±  a  -  b  V^Hf )  (a?  =fc  a  +  6  V^^T)  =  a?  ±  2ax  +  a2  +  b2. 

Hence,  for  each  set  of  imaginary  roots  which  arise  from 
placing  the  denominator  of  the  fraction  equal  to  0,  there 
will  be  a  factor  of  the  second  degree  of  the  form 

x2  ±  2ax  +  a2  +  b2. 
257.  If  the  imaginary  roots  are  equal,  we  shall  have, 

«  =  0,     x  =  +  b  V  —  1 ,     x——  bi/^~I, 

and  the  factor  will  become     a?  -f  b2. 
In  the  equation, 

a?  -6c#+  lOc^O, 
the  roots  are, 

x  =  3c  +  c  V—  1,     #  =  3c  —  c  V—  1 ; 

comparing  these  values  of  x  with  the  general  form,  we 
have 

a  =  —  3c          6  —  c, 

and  the  given  equation  takes  the  form 

a?  —  6cx-\-9c2-\-c2  =  Q. 
Comparing  the  roots  of  the  equation, 

0^-1-4^+12  =  0, 
with  the  values  of  x  in  the  general  form,  we  have 


228  ELEMENTS    OF    THE 

and  the  equation  may  be  written  under  the  form 


258.  Let  us  first  consider  the  case  in  which  the  deno- 
minator of  the  fraction  to  be  integrated  contains  but  one 
set  of  imaginary  roots.  The  fraction  will  then  be  of  the 
form, 

_  P+Qx  +  Ra?-\-Sa?-\-  &c.  __  , 
(x-a)(x-b)  ----  (x  -  h)  (a?+2ax  +  a2  +  P)     ' 

which  may  be  placed  under  the  form 

Adx       Bdx  Hdx  Mx  +  N 

' 


x  —  a      x  —  b'  x  —  h 

The  first  three  fractions  may  be  integrated  by  the  methods 
already  explained  :  it  therefore  only  remains  to  integrate 
the  last,  which  may  be  written  under  the  form 

Mx  +  N 


If  we  make    x  +  a  =  z,     the  expression  becomes 

Mz  +  N-Ma  , 
z2  +  b2       dz> 

and  making     N  —  Ma  —  P,     it  reduces  to 

Mz+P 


which  may  be  divided  into  the  parts, 
Mzdz  ,     Pdz 

Z2+tf  +  z2  + 

which  may  be  integrated  separately. 


INTEGRAL    CALCULUS.  229 

To  integrate  the  first  term,  we  have 

Mzdz       ,     C  zdz        M  r  2zdz 


in  which  the  numerator,  2zdz,  is  equal  to  the  differential 
of  the  denominator:  hence  (Art.  218), 

CMzd*__M_lQ  (z*  +  b2)' 

or  by  substituting  for  z  its  value,     x  +  a, 
*Mzdz      M 


=  M  log  yV  +  2ax  +  a2  +  i2. 
Integrating  the  second  term  by  Art.  224,  gives 


or  by  substituting  for  z  its  value,     x  -\-  a,     and  for  P, 
N  —  Ma,     we  have 


.=___tang. 

and  finally, 

Af  log  VV  +  2ax  +  a2  +  b2  -\ ^— —  tang"1  (~ ^~)- 

259.  Let  us  take,  as  an  example,  the  fraction 


230  ELEMENTS    OF    THE 

in  which,  if  +1  be  substituted  for  a?,  the  denominator 
will  reduce  to  0 :  hence,  x  —  1  is  a  factor  of  the  denomi- 
nator. Dividing  by  this  factor,  the  fraction  may  be  put 
under  the  form 

.; :  ;    .__^+£___(ir,     -  -  iv 

in  which  x2  -f  x  -+-  I  is  the  product  of  the  imaginary 
factors.  Placing  this  product  equal  to  0,  finding  the  roots 
of  the  equation,  and  comparing  them  with  the  general 
values  in  the  form 


<?  +  b*=0, 
we  find 

1  /T 
a  —  —         6  —  \/  — . 

2  V    4 

We  may  place  the  given  fraction  under  the  form 

c  +  fx  _     A  MX  +  N 

(#-r  '  a  '       ~~ 


reducing  the  second  member  to  a  common  denominator, 
and  comparing  the  coefficients  of  x  in  the  numerator  with 
those  of  x  in  the  numerator  of  the  first  member,  we  obtain 


Substituting  these  values  of  M  and  N,  as  also  those  of  a 
and  b,  in  the  general  formula  of  Art.  258,  and  recollecting 
that 

r  Adx       c+f   r   dx         c+f, 
I  -  =  —  -^-  I  -  =  —  —logix  —  1  ), 
J  x-l  3    J  x-l          3 


INTEGRAL    CALCULUS.  231 

we  find 


260.  The  equation  which  arises  from  placing  the  de- 
nominator of  the  fraction  equal  to  0,  may  give  several 
pairs  of  imaginary  roots  respectively  equal  to  each  other. 
In  this  case,  the  factor  x2  ±  2ax  +  a2  +  b2  will  enter 
several  times  into  the  denominator,  or  will  take  the  form 


and  hence,  that  part  of  the  fraction  which  contains  the 
pairs  of  equal  and  imaginary  roots,  must  be  placed  under 
the  form  (Art.  251) 

H+Kx  H  +  K'x 


2ax  +  a  p~ 


H"  +  K"x  Hn  +  Knx 

"*" 


(x2  +  2ax  +  a2  -f  62)p~s  or2  +  2aa?-+  «2  +  &2' 

Now,  reducing  to  a  common  denominator,  and  comparing 
the  coefficients,  we  find  the  values  of  the.  constants 

H,    K,    H,    K',    H",    K" Hn,    Kn . .  ^ 

after  which,  multiply  each  term  by  dx,  and  then  integrate 
the  terms  separately. 

Since  all  the  terms  are  of  the  same  general  form,  it  will 
only  be  necessary  to  integrate  the  first  term,  which  may 
be  written  under  the  form 

H+  Kx 

dx; 


232  ELEMENTS    OF    THE 

which,  if  we  make  x  +  a  =  zr  will  reduce  to 
H-Ka+_ Kz 

(#+&Y  ' 

and  making  M—H—  Ka,  it  will  become 

M+Kz    .  Kzdz  Mdz 


The  first  term  of  the  second  member  may  be  placed  under 
the  form 


and  integrating  by  the  formula  of  Art.  217,  we  have 
Kzdz          1       K  I 


2  (l- 
It  then  only  remains  to  integrate  the  second  term 


V     r~.j 

By  comparing  the  second  member  of  this  equation  with 
formula  (D),  Art.  242,  we  see  that  it  will  become  identical 
with  the  first  member  of  that  formula,  by  supposing 

m=l,       a  =  b2,       b  =  l,       and       ft  =  2; 

and  hence,  by  means  of  that  formula,  the  exponent  —  p 
may  be  successively  diminished  by  unity  until  it  becomes 
—  1,  when  the  integration  of  the  term  will  depend  on 
that  of 


But  we  have  already  found  (Art.  224), 
dz2          1 


i/z\ 

(T; 


INTEGRAL    CALCULUS.  233 

and  hence  the  fraction  may  be  considered  as  entirely  in- 
tegrated. 

261.  It  follows,  from  the  preceding  discussion,  that  the 
integration  of  all  rational  fractions  depends  on  the  follow- 
ing forms  : 

1st.        fxmdx  = 


2d.  .  =  ±  log(a  ±  a?). 

•/  a  i  x 

C    dx          1  \fx\ 

3d.  /    2  ,     *  =  —  tang     (  —  V 

J  a2  +  x2      a  \aJ 


Integration  of  Irrational  Fractions. 

262.  The  method  of  integrating  rational  fractions  having 
been  explained,  we  may  consider  an  irrational  fraction  as 
admitting  of  integration  when  it  is  reduced  to  a  rational 
form. 

263.  Every   irrational   fraction   in   which    the   radical 
quantities  are  monomials,  may  be  reduced  to  a  rational 
form. 

Let  us  take,  as  an  example, 


or 


Having  found  the  least  common  multiple  of  the  indices 
of  the  roots,  (which  indices  are  the  denominators  of  the 
fractional  exponents,)  substitute  for  x  a  new  variable,  z, 
with  this  common  multiple  for  an  exponent,  and  the  frac- 
tion will  then  become  rational  in  terms  of  z. 


234  ELEMENTS    OF    THE 

In  the  example  given,  the  least  common  multiple  is  6; 
hence  we  have 


x  —  z6     and 
and  substituting  these  values,  we  obtain 

^-fr-^zi^^g^*!^. 

V  a?  —  v  a?     z  ' 

an  expression  which  may  be  integrated  by  rational  frac- 
tions ;  after  which  we  may  substitute  for  z  its  value,   -\/x. 

264.  If  the  quantity  under  the  radical  sign  is  a  polyno- 
mial, the  fraction  cannot,  in  general,  be  reduced  to  a 
rational  form.  We  can,  however,  reduce  to  a  rational 
form  every  expression  of  the  form 


in  which  X  is  supposed  to  be  a  rational  function  of  x. 

If  we  write  a  denominator  1,  and  then  multiply  the 
numerator  and  denominator  by  VA  -f  Bx  ±  Co?2,  the 
expression  will  take  the  form 

Xfdx 


in  which  XI  is  a  rational  function  of  x:  hence  the  two 
forms  are  essentially  the  same. 

If  now,  we  can  find  rational  values  for  V  A  +  Bx  i  Cx2 
and  for  dx,  in  terms  of  a  new  variable,  the  expression  will 
take  a  rational  form. 

There  are  two  cases  to  be  considered:  1st.,  when  tb» 
coefficient  of  x2  is  positive  ;  and,  2d,  when  it  is  negative 


INTEGRAL    CALCULUS.  235 

Let  us  consider  them  separately.     First,  make 


=  VC  Va  +  bx  +  x2, 

A      ,      B 

m  which     a  =  —-,     b  =  -—. 

O  G  

In  order  to  find  rational  values  for  dx  and  Va  +  bx+x2, 
place 

Va  +  bx  +  x2  =  x  +  *,     (1) 
from  which,  by  squaring  both  members,  we  find 

a  +  ta  =  2#z  +  z2,     (2) 
and  hence, 

i^SSt^M^   (3)  ; 

and  substituting  this  value  in  equation  (1), 


y  a  -f  bx  +  x2  — h  %  5 

6-2* 

and  by  reducing  to  the  same  denominator, 

z2  —  bz  +  a 


Let  us  now  find  the  value  of  dx  in  terms  of  z.     For  this 
purpose  we  will  differentiate  equation  (2),  we  then  find 

bdx  =  2xdz  +  2zdx  +  2zdz  ; 
whence  we  have 

(b  —  2z)dx  —  2(x  +  z)dz ; 


236  ELEMENTS    OF    THE 

and  by  subtracting  equations  (1)  and  (4),  and  substituting 
for  x  +  z  the  value  thus  found,  we  have 


265.  Let  us  take,  as  an  example, 
dx 


x     A  +  Bx  +  Co? 

which  may  be  written  under  the  form 

dx 


V~C  x  x 

and  substituting  the  values  of  Va  +  bx  +  x2  and  dx,  from 
equations  (4)  and  (5),  we  have 

dx  2dz 


and  multiplying  the  denominator  by  the  value  of  a?,  in 
equation  (3), 

dx  2dz 


and  then  by  Vc,  we  have 


dx  dx  2dz 

:,  or 


X  x  Va  +  bx  +  x VA+Bx+Cx*   (z2-« 

which  is  a  rational  form,  and  may  be  integrated  by  the 
methods  already  explained. 


INTEGRAL    CALCULUS.  237 

266.  Let  us  take,  as  a  second  example, 
dx 


which  may  be  placed  under  the  form 

dx 


and  comparing  this  with  the  form  of  Art.  264,  gives 


Hence, 

/dx  I     s*     dx 

Vft  +  cV~~~  J  VoT^ 


Having  placed 

Va  +  a?2  —  z  +  x, 

we  found,  Art.  264,  equations  (5)  and  (4), 

a 


hence 

/c?a?  /•'    efe          , 

.  =    /  --  =—  log  2:. 

Va  +  a?      J        ^ 

Substituting  for  z  its  value,  and  multiplying  by    —  ,      we 
have 


and  substituting  for  a  its  value,     —  ,     we  have 


238 


ELEMENTS    OF    THE 


ll  "1 

=  -7logl_7 


=  _  —log—  -  —  log(  Vh  +  <?a*  -  ex)  +  C. 
c         c        c 

But  since  the  difference  of  the  squares  of  the  tvVo  terms 
within  the  parenthesis  is  equal  to  A,  it  follows  that  if  h 
be  divided  by  the  difference  of  the  terms,  the  quotient  will 
be  their  sum  (Alg.  Art.  59).  But  the  division  may  be 
effected  by  subtracting  their  logarithms.  Let  us,  then, 
add  to,  and  subtract  from,  the  second  member  of  the  equa- 
tion, — log  h.  We  shall  then  have, 


or  by  representing  the  three  constants  — log 

C  CO 

and  C,  by  a  single  letter  C,  we  have 

=  — log(  Vh  +  cV  +  ex)  -J-  C. 


267.  Let  us  take,  as  a  third  example, 


Comparing  this  with  the  general  form,  we  find 

a  =  mz    and    6  =  0; 
nence  (Art.  264), 

*2  (z*  +  m2 

and     •fr^*&Z— 


INTEGRAL    CALCULUS.  239 

and  consequently, 

/    8  ,     a 

dx  V  m  +  x2  =  — 


which  is  rational  in  z  ;  and,  having  found  the  integral  in  z, 
substitute  the  value  of  z  in  terms  of  x. 

268.  Let  us  now  consider  the  case  in  which  the  coeffi- 
cient of  x2  is  negative.     We  have 


If  now,  we  make  as  before, 


Va  4-  bx  —  o?  —  x-\-zy 

and  square  both  members,  the  second  powers  of  x  in  each 
member  will  not  cancel,  as  before  ;  and  therefore,  x  can- 
Aot  be  expressed  rationally  in  terms  of  z.  We  must, 
therefore,  place  the  value  of  the  radical  under  another 
form.  We  will  remark,  in  the  first  place,  that  the  bino- 
mial a  +  bx  —  a?,  may  be  decomposed  into  two  rational 
factors  of  the  first  degree,  with  respect  to  x.  For,  if  we 

make 

x2  —  bx  —  a  —  Q, 

and  designate  the  roots  of  the  equation  by  *  and  «',  we 
have  (Alg.  Art.  142) 


and  consequently,  by  changing  the  signs, 

-x*)=-(x-«)(x-«f)  =  (x 
16 


240  ELEMENTS    OF    THE 

and  placing  the  second  member  under  the  radical,  we 
may  make 

y  (x  _  *)  (  *'  -  a?)  =  (a?  -  «)  z  ;     (1) 

squaring  both  members 


and  by  suppressing  the  common  factor  x  —  *, 

«'_tf:=(#-«)z2,       (2) 

whence, 

*!  +  *z2 

*=TT?-' 

*'  +  *z2 
and  a?-<*  =  -'-" 

or  by  reducing, 


which,  being  substituted  in  the  second  member  of  equa- 
tion (1),  gives 

=  -*;     (4) 


and  by  differentiating  equation  (3),  we  obtain 


269.  To  apply  this  method  to  a  particular  example  of 

the  form 

dx 


INTEGRAL    CALCULUS.  241 

substitute  the  values  of  Va  +  bx  —  x2  and  dx,  found  in 
equations  (41  and  (5) :  we  find 

dx  2(*'  —  *)z  2dz 

• — : (LZ  = 


a       x-x  (l, 

hence 


/d°°  _       o         -i     ,   p . 

Va  +  bx  —  x2 

or,  by  substituting  for  z  its  value  from  equation  (1), 
dx 


f         x        =r:- 

J  Va  +  bx-a? 


270.  If,  in  the  last  formula,  we  make 
a  =  1     and     6  =  0, 

the  trinomial  under  the  radical  will  become   I  —x2,  and 
the  roots  of  the  equation  x2  —  1  =  0  are 

«-  =  —  I     and     *'  =  I . 
Substituting  these  values,  and  the  general  formula  becomes 

/dx  ~  _t      /I  — x 

Vl  —  x2  V    l  -{-# 

and  if  we  suppose  the  integral  to  be  0  when  x  =  0,  we 
shall  have 

0=C-2tang-1(l) 
=  C  -  2(45°)      (Trig.  Art.  VIII) 


=  C-90°:      hence     C  =  —  . 


242  ELEMENTS    OF    THE 

Substituting  this  value,  and  we  have 
dx 


x  5r 

7CT=" 


271.  We  have  already  seen  (Art.  219)  that 

/dx  .      . 

—  ===  =  sm  lx\ 
Vl-a? 

and  hence, 


should  also  represent  the  arc  of  which  x  is  the  sine 
To  prove  this,  we  have  (Trig.  Art.' XXV) 


2  tang  A 
1  —  tang2A 


/I—a? 

Substituting  for  tang  A,  v/  ,  and  reducing,  we  have 

v    \-\-x 


that  is,  twice  the  arc  whose  tangent  is    \J  - — -    is  equal 

"    1  +  a? 

to  the  arc  whose  tangent  is     ^  - 

x 

But  the  arc  whose  tangent  is  T     is  the  com- 

x 

M 

plement  of  the  arc  whose  tangent  is        ,          -,     (Trig. 

*V  1  —  <it 

Art.  XVIII);  and  this  arc  has  x  for  its  sine.     Hence, 
either  member  of  the  equation 


INTEGRAL    CALCULUS.  243 

dx  ^  /I  — a? 


/_!          /I 
vi=?-~2~~         Vr+J 

represents  the  arc  whose  sign  is  x. 

272.  Let  us  take,  as  a  last  example,  the  differential 


—  x2. 

In  comparing  this  with  the  general  form,  we  find  (Art 
268) 

*  =  Q     and     «!  =  2a; 


and  Art.  268,  equations  (4)  and  (5),  give 
i—r^ r        2az 


dz. 


Substituting  these  values,  we  have 


which   may   be   integrated    by   the    method   of   rational 
fractions. 

Rectification  of  Plane  Curves. 

273.  The  rectification  of  a  curve  is  the  expression  for 
its  length.     When  this  expression  can  be  found  in  a  finite 
number  of  algebraic  terms,  the  curve  is  said  to  be  rectifiable, 
and  its  length  may  be  represented  by  a  straight  line. 

274.  The  differential  of  the  arc  of  a  curve,  referred  to 
rectangular  co-ordinates,  is  (Art.  128) 


dz  = 


244  ELEMENTS    OF   THE 

Hence,  if  it  be  required  to  rectify  a  curve,  given  by  its 
equation, 

1  st.  Differentiate  the  equation  of  the  curve. 

2d.  Combine  the  differential  equation  thus  found  with 
the  given  equation,  and  find  the  value  of  dx2  or  dy2  in 
terms  of  the  other  variable  and  its  differential. 

3d.  Substitute  the  value  thus  found  in  the  differential 
of  the  arc,  which  will  then  involve  but  one  variable  and 
its  differential.  Then,  by  integrating,  we  shall  find  an 
expression  for  the  length  of  the  arc,  estimated  from  a 
given  point,  in  terms  of  one  of  the  co-ordinates. 

275.  Let  us  take,  as  a  first  example,  the  common  para- 
bola, of  which  the  equation  is 

y2  =  2px. 
Differentiating,  and  dividing  by  2,  we  have 

ydy 
and  consequently, 


substituting  this  value  in  the  differential  of  the  arc,  we 
have 


which,  being  integrated  by  formula  (B)  Art.  239,  gives 
by  supposing    m  =  \,    a=p",    6=1,     n  =  2,    P=-> 


INTEGRAL    CALCULUS. 


245 


and  integrating  the  second  term  by  the  formula  of  Art. 
266,  we  have,  after  maKing  h=p\  c2=l, 


and  consequently, 


If  we  estimate  the  arc  from  the  vertex  of  the  parabola, 
we  shall  have 

y  =  0     for    z  =  0  :     hence 

C  or     C=— 


and  consequently, 


and  hence,  the  value  of  the  arc,  for  a  given  ordinate  y,  can 
only  be  found  approximative^. 

276.  The  curves  represented  by  the  equation 


are  called  parabolas     This  equation  may  be  placed  under 
the  form 


or  by  placing    pn=pf,    and      —  =  nf,    we  have 


246  ELEMENTS    OF    THE 

or  finally,  by  omitting  the  accents,  the  form  becomes 

y=pxn. 
By  differentiating,  we  have 

dy  =  npxn~ldx, 

and  by  substituting  this  value  of  dy  in  the  differential  of 
the  arc,  we  have 


The  integral  of  this  expression  will  be  expressed  in  a 
finite  number  of  algebraic  terms  when  -  -  -    is   a  whole 

number  and  positive  (Art.  235),  If  we  designate  such 
whole  and  positive  number  by  i,  we  have  for  the  condition 
of  an  exact  integral  in  algebraic  terms, 

1  2i+l 


and  substituting  for  n,  we  have 


which  expresses  the  relation  between  x  and  y  when  the 
length  of  the  arc  can  be  found  in  finite  algebraic  terms. 
There  is  yet  another  case  in  which  the  integral  will  be  ex- 

pressed in  finite  and  algebraic  terms,  viz.  when  -   —  ^+TT 

liTi  -  £  £ 

is  a  positive  whole  number  (Art.  236  and  235.) 

o 

277.  If  we  make  i  —  1,  we  have  n  =  —  ,  and 


which  is  the  equation  of  the  cubic  parabola. 


INTEGRAL    CALCULUS.  247 

Under  this  supposition,  the  arc  becomes  (Art.  217) 

i  +      »)*  f  c  ; 


and  hence,  the  cubic  parabola  is  rectifiable  (Art.  273). 

If  we  estimate  the  arc  from  the  vertex  of  the  curve,  we 
have     x  =  0,     for    z  =  0  :     hence 


or        =- 


and  consequently, 


z  = 


278.  If  the  origin  of  co-ordinates  is  at  the  centre  of  tne 
circle,  the  equation  of  the  circumference  is 


and  the  value  of  the  arc, 


If  the  origin  be  placed  on  the  curve 
y2  =  2Rx  —  a?, 
dx 


and 


both  of  which  expressions  may  be  integrated  by  series, 
and  the  length  of  the  arc  found  approximatively. 

279.  It  remains  to  rectify  the  transcendental  curves. 
The  differential  equation  of  the  cycloid  is  (Art.  182) 


7 

dx  =  —  , 

V  2ry  —  y2 


ELEMENTS    OF    THE 


248 
which  gives 


Substituting  this  value  of  da?  in  the  differential  of  the 
arc,  we  obtain 


2ry-y* 


_ 
=  (2rY(2r-y) 


But  (Art.  217) 


and  hence, 


If  now,  we   estimate 
the  arc  z  from  B,  the 
point  at  which  y  =  2r, 
we  shall  have,  for  z  =  0,     \ 
y  =  2  r ;   hence  -A      F 

0  =  0  +  C,     or     C  =  0, 

and  consequently,  the  true  integral  will  be 


the  second  member  being  negative,  since  the  arc  is  a 
decreasing  function  of  the  ordinate  y  (Art.  31). 

If  now,  we  suppose  y  to  decrease  until  it  becomes 
equal  to  any  ordinate,  as  DF  =  ME,  DB  will  be  repre- 
sented by  z,  or  by  2  -y/2r(2r  —  y\  and  BE  =  2r  —  y. 

But    5G2  =  BM x  BE :     hence 


INTEGRAL    CALCULUS. 

and  consequently 


or  the  arc  of  the  cycloid,  estimated  from  the  vertex  of  the 
axis,  is  equal  to  twice  the  corresponding  chord  of  the 
generating  circle  :  hence,  the  arc  BDA  is  equal  to  twice 
the  diameter  BM  ;  and  the  curve  ADBL  is  equal  to  four 
times  the  diameter  of  the  generating  circle. 

280.  The  differential  of  the  arc  of  a  spiral,  referred  to 
polar  co-ordinates,  is  (Art.  202) 


Taking  the  general  equation  of  the  spirals 
u  -  at", 

we  have  du2  =  nWn-W', 

and  substituting  for  du2  and  u2  their  values,  we  obtain 


If  we  make  n  =  l,  we  have  the  spiral  of  Archimedes, 
(Art.  191),  and  the  equation  becomes    . 

dz  =  adtVl  -I-*2; 

which  is  of  the  same  form  as  that  of  the  arc  of  the  com- 
mon parabola  (Art.  275). 

281.  In  the  logarithmic  spiral,  we  have  i  =  logw,  and 
the  differential  of  the  arc  becomes 

dz  =  duV2+C; 
and  if  we  estimate  the  arc  from  the  pole, 


250  ELEMENTS    OF    THE 

Consequently,  the  length  of  the  arc  estimated  from  the 
pole  to  any  point  of  the  curve,  is  equal  to  the  diagonal  of 
a  square  described  on  the  radius-vector,  although  the 
number  of  revolutions  of  the  radius-vector  between  these 
two  points  is  infinite. 

Of  the  Quadrature,  of  Curves. 

282.  The  quadrature  of  a  curve  is  the  expression  of  its 
area.     When  this  expression  can  be  found  in  finite  alge- 
braic terms,  the  curve  is  said  to  be  quadrable,  and  may  be 
represented  by  an  equivalent  square. 

283.  If  5  represents  the  area  of  the  segment  of  a  curve, 
and  x  and  y  the  co-ordinates  of  any  point,  we  have  seen 
(Art.  130), -that 

ds  =  ydx. 

To  apply  this  formula  to  a  given  curve  : 

1st.  Find  from  the  equation  of  the  curve  the  value  of  y 
in  terms  of  x,  or  the  value  of  dx  in  terms  of  y,  which 
values  will  be  expressed  under  the  forms 

y  =  f(x\     or     dx=f(y)dy. 

2d.  Substitute  the  value  of  y,  or  the  value  of  dx,  in  the 
differential  of  the  area  :  we  shall  have 

ds  =  /  (x)  dx,     or     ds  =  f  (y}  dy : 

the  integral  of  the  first  form  will  give  the  area  of  the 
curve  in  terms  of  the  abscissa,  and  the  integral  of  the 
second  will  give  the  area  in  terms  of  the  ordinate. 


INTEGRAL    CALCULUS  251 

284.  Let  us  take,  as  a  first  example,  the  family  of  para- 
bolas of  which  the  equation  is 


we  shall  then  have 

and 

—  -  no"     **'*'*          n 

fF(x)dx=fpnxndx=—^-—x  n    = — — -xy+C; 

1.  — 
by  substituting  y  for  its  value,    pnxn. 

If,  instead  of  substituting  the  value  of  y  in  the  differential 
of  the  area 

ydx, 

we  find  the  value  of  dx  from  the  equation 
we  have 


and  consequently, 


by  substituting  x  for  its  value,    —  7,  which  is  the  same  re- 

pm 

suit  as  before  found. 

Hence,  the  area  of  any  portion  of  a  parabola  is  equal 
to  the  rectangle  described  on  the  abscissa  and  ordinate 


252  ELEMENTS    OF    THE 

multiplied  by  the  ratio     -  .     The  parabolas  are  there- 
m  +  n 

fore  quadrable. 

In  the   common  parabola,     n  =  2,     m=l,     and  we 
have 


=  —  xy, 

that  is,  the  area  of  a  segment  is  equal  to  two  thirds  of 
the  area  of  the  rectangle  described  on  the  abscissa  and 
ordinate. 

285.  If,  in  the  equation 

f-jf, 

we  make  n  —  1,  and  m  —  1,  it  will  represent  a  straight 
line  passing  through  the  origin  of  co-ordinates,  and  we 
shall  have 

Jf(x)dx  =  —  xy, 

which  proves  that  the  area  of  a  triangle  is  equal  to  half 
the  product  of  the  base  and  perpendicular. 

286.  It  is  frequently  necessary  to  find  the  integral  or 
1  unction,  between  certain  limits  of  the  variable  on  which 
it  depends. 

A  particular  notation  has  been  adopted  to  express  such 
integrals. 

Resuming  the  equation  of  the  common  parabola 


and  substituting  in  the  equation  ydx  the  value  of  dx  ~    -, 
we  have 


INTEGRAL    CALCULUS.  253 


or,  •'    the  area  be  estimated  from  the 
vr/tex  A,  we  have  C  =  0,  and 


3P 

If  now,  we  wish  the  area  to  terminate      J 

at  any  ordinate  PM—  6,  we  shall  then 

take  the  integral  between  the  limits  of  y  =  0  and  y  =  b; 

and,  to  express  that  in  the  differential  equation,  we  write 


which  is  read,  integral  of  y*dy  between  the  limits  y  =  0 
and  y  =  b. 

If  we  wish  the  area  between  the  ordinates  MP  =  b, 
MP'  =  c,  we  must  integrate  between  the  limits  y  =  6, 
y  —  c.  We  first  integrate  between  0  and  each  limit,  viz. : 

1    rb    2,         63 
'  pj  0  ;     y      3p' 


we  then  have 


PMM'P  =  AMM'P'  -  AMP  =  —  C  °  yzdy 


-- 

3p       3p       3p  v 

287.  Let  us  now  determine  the  area  of  any  portion  of 
the  space  included  between  the  asymptotes  and  curve  of 
an  hyperbola. 


254 


ELEMENTS    OF    THE 


The  equation  of  the  hyperbola  referred  to  its  asymp- 
totes (An.  Geom.  Bk.  VI,  Prop.  IX,)  is 

xy  —  M. 

In  the  differential  of  the  area  of  a  curve  ydx,  x  and  y 
are  estimated  in  parallels  to  co-ordinate  axes,  at  right  an- 
gles to  each  other. 

The  differential  of  the 
area  BCMP,  referred  to 
the  oblique  axes  AX, 
A  Y,  is  the  parallelogram 
PMMP',  of  which 
PM=y  and  PP'  =  dx. 

If   we   designate   the 
angle  YAX  =  MPX  by 
/3,  we  shall  have 

area  PMM'P  =  ydxsmp  ; 

and  substituting  for  y  its  value    —  ,     and  representing 

x 

the  area  BCMP  by  s,  we  have 

,       ,-  .     dx 

ds  =  Msir\&  —  , 
x 

/dx 
—  =  M  sin  /slog  a?  +  C. 

If  AC  is  the  semi-transverse  axis  of  the  hyperbola,  and  we 
•make  AB=l,  and  estimate  the  area  s  from  J5C,  we  shall 
have,  for  x  —  1  ,  5  =  0,  and  consequently  C  =  0  ;  and  the 
true  integral  will  be 

s  = 


INTEGRAL    CALCULUS.  255 

But,  since  ABCD  is  a  rhombus,  and  M  =  AB  x  BC  (An. 
Geom.  Bk.  VI,  Prop.  IX,  Sch.  2),  and  since  AB  =  1,  we 
have  M=l,  and  consequently, 

s  =  sin  /slog  x. 

Now,  since  s,  which  represents  the  space  BCMP  for.  any 
abscissa  a?,  is  equal  to  the  Naperian  logarithm  of  x  multi- 
plied by  the  constant  sin/3,  s  may  be  regarded  as  the  loga- 
rithm of  x  taken  in  a  system  of  which  sin(S  is  the  modu- 
lus (Alg.  Art.  268).  Therefore,  any  hyperbolic  space 
BCMP  is  the  logarithm  of  the  corresponding  abscissa 
AP,  taken  in  the  system  whose  modulus  is  the  sine  of  the 
angle  included  between  the  asymptotes. 

If  we  would  make  the  spaces  the  Naperian  logarithms 
of  the  corresponding  abscissas,  we  make  sin/3  =  1,  which 
corresponds  to  the  equilateral  hyperbola.  If  we  would 
make  the  spaces  the  common  logarithms  of  the  abscissas, 
make  sin/3  =,  0.43429945,  (Alg.  Art.  272). 

288.  The  equation  of  the  circle,  when  the  origin  of  co- 
ordinates is  placed  on  the  circumference,  is 


t/2  =  2rx  —  x2,     or     y 
and  hence,  the  differential  of  the  area  is 


dx  v*2rx  —  x2  ; 
and  this  will  become,  by  making    x  =  r  —  u, 


If  we  integrate  this  expression  bv  formula  (B,  Art.  239, 

17 

I 


256  ELEMENTS    OF    THE 

we  have 

" 


2 
But  we  have  (Art.  253) 

-du 


/—u 
.-* s  = 
yr*  —  u 

and  placing  for  u  its  value 


— 
2 

and  taking  this  integral  between  the  limits  a?  =  0  and 
#  =  2r,  we  shall  have  the  area  of  a  semicircle. 

For  07  =  0,  the  area  which  is  expressed  in  the  first 
member  becomes  0,  the  first  term  in  the  second  member 
becomes  0,  and  the  second  term  also  becomes  0,  since 
the  arc  whose  cosine  is  1,  is  0:  hence  the  constant 
C  =  0. 

If  we  now  make     x  =  2r,     the  term 


—  (r  —  x)  -/  2rx  —  of 
2 

reduces  to  0,  and  the  second  term  to 

-i-r'cos-X-  1)  =  — r3*1    (Trig.  Art.  XIV), 
and  consequently,  the  entire  area  is  equal  to    r2^,    which 


INTEGRAL    CALCULUS.  257 

corresponds  with  a  known  result  (Geom.  Bk.  V,  Prop.  XII, 
Cor.  2). 

The  equation  of  the  ellipse,  the  origin  of  co-ordi- 
nates being  at  the  vertex  of  the  transverse  axis  (An.  Geom. 
Bk.  IV,  Prop.  I.  Sch.  8),  gives 


B 

y=A 

and  consequently,  the  area   of  the   semi-ellipse  will,  be 
equal  to 


T>          /• 

—  —   I 


Integrating,  as  in  the  last  example,  between  the  limits 
x  =  0,  and  x  —  2A,  and  multiplying  by  2,  we  find  AB* 
for  the  entire  area.  This  corresponds  with  a  known  result 
(An.  Geom.  Bk.  IV,  Prop.  XIII). 

289.  The  differential  equation  of  the  cycloid  (Art.  183)  is 


whence 


and  applying  formula  E,  (Art.  243)  twice,  it  will  reduce  to 

-;  and  (Art.  226) 


But  we  may  determine  the  area  of  the  cycloid  in  a  more 
simple  manner  by  introducing  the  exterior  segment  AFKH, 


258 


ELEMENTS    OP   THE 


Regarding  FB   as   a       F     K 
line  of  abscissas,  and  de- 
signating any  ordinate  as 
KH,  by  z  =  2r  —  y,  we 
shall  have 


B 


But 


whence 


d(AFKH)  = 


zdx  = 


, 
V2ry  —  y2 


But  this  integral  expresses  the  area  of  the  segment  of  a 
circle,  of  which  the  abscissa  is  y  and  radius  r  (Art.  288): 
that  is,  of  the  segment  MIGE.  If  now,  we  estimate  the 
area  of  the  segment  from  M,  where  y  =  0,  and  the  area 
AFKH  from  AF,  in  which  case  the  area  AFKH=  0  for 
y  ;=  0,  we  shall  have 

AFKH  =  MIGE; 

and  taking  the  integral  between  the  limits  y  =  0  and 
y  =  2r,  we  have 

AFB  =  semicircle  MIGB, 
and  consequently, 

area  AHBM  =  A  FBM  -  MIGB. 

But  the  base  of  the  rectangle  AFBM  is  equal  to  the  semi- 
circumference  of  the  generating  circle,  and  the  altitude  is 
equal  to  the  diameter,  hence  its  area  is  equal  to  four  times 
the  area  of  the  semicircle  MIGB  ;  therefore, 


area  AHBM  =3  MIGB, 


INTEGRAL    CALCULUS.  259 

and  consequently,  the  area  AHBL  is  equal  to  three  times 
the  area  of  the  generating  circle. 

290.  It  now  remains  to  determine  the  area  of  the  spirals. 
If  we  represent  by  s  the  area  described  by  the  radius-vec- 
tor, we  have  (Art.  203) 

u2dt 

'  dS  =  —  ; 

and  placing  for  u  its  value  atn  (Art.  189) 

~2,2n  7,  A/2/2n  +  l 

-.         a  i    (11  ,  u  L  I    /-i 

ds  =  -     and     s  =  -  -  —  -  +  C, 
2  4n  +  2 

and  if  n  is  positive  C  =  0,  since  the  area  is  0  when  t  =  0. 
After  one  revolution  of  the  radius-vector,  t  —  2  T,  and  we 
have 

= 


471  +  2 

which  is  the  area  included  within  the  first  spire. 
291.  In  the  spiral  of  Archimedes  (Art.  192) 

a  =.  —     and     n  =  1  ; 
2?r 

hence,  for  this  spiral  we  have 

*3 
= 


which  becomes    ~,    after  one  revolution  of  the  radius- 

7T 

vector  ;  the  unit  of  the  number  —  being  a  square  whose 

3 

side  is  unity.  Hence,  the  area  included  by  the  first  spire, 
is  equal  to  one  third  the  area  of  the  circle  whose  radius  is 
equal  to  the  radius-vector  after  the  first  revolution. 

In  the  second  revolution,  the  radius-vector  describes  a 


260  ELEMENTS    OF    THE 

second  time  the  area  described  in  the  first  revolution  ;  and 
m  any  revolution,  it  will  pass  over,  or  redescribe,  all  the 
area  before  generated.  Hence,  to  find  the  area  at  the  end 
of  the  mth  revolution,  we  must  integrate  between  the  limits 

t  =  (m—  1)2^     and     t  =  m.2vr, 
which  gives 


3 

If  it  be  required  to  find  the  area  between  any  two  spires, 
as  between  the  mth  and  the  (m  -f  1  )th,  we  have  for  the 
whole  area  to  the  (m  +  l)th  spire  equal  to 
(m-fl)3-m3 

~3~ 
and  subtracting  the  area  to  the  mth  spire,  gives 


o 

for  the  area  between  the  mth  and  (m  -f  l)th  spires. 

If  we  make  m  —  1,  we  shall  have  the  area  between  the 
first  and  second  spires  equal  to  2?r:  hence,  the  area  be- 
tween the  mth  and  (m+  l)th  spires,  is  equal  to  m  times 
the  area  between  the  first  and  second. 

292.  In  the  hyperbolic  spiral  n  =  —  1,  and  we  have 

ds  =  -  dt     and     s  =  --  . 
2  2t 

The  area  s  will  be  infinite  when  t  =  0,  but  we  can  find 
the  area  included  between  any  two  radius-vectors  b  and  c 
by  integrating  between  the  limits  t  =  b,  t  =  c,  which  will 

give 

_  1 

S~ 


INTEGRAL    CALCULUS.  261 

293.  In  the  logarithmic  spiral  t  =  logw  :  hence,  dt  =  — , 
u?dt       udu 


hence, 


6'  = 


2  2 

udu  _uz 
~=~  L; 


and  by  considering  the  area  s  =  Q  when  u  =  0,  we  have 
C  =  0  and 


Determination  of  the  Area  of  Surfaces  of 
Revolution. 


294.    If  any  curve  EMM,  be   re- 
volved about  an  axis  AX,  it  will  de- 
scribe   a    surface    of    revolution,    and 
every  plane  passing  through  the   axis 
AX  will  intersect  the  surface  in  a  me-    j 
ridian  curve.     It  is  required  to  find  the 
differential  of  this  surface.      For  this      A        P 
purpose,  make  AP  —  x,  PM  =  y,  and  PPf  —  h  :  we  shall 
then  have 


/ 

B 

PM  =  f(x)  =  y, 

P'M'  =  f(x  +  h)  =  y  +  ^-, 


-  +  &c. 


262 


ELEMENTS    OF    THE 


In  the  revolution  of  the  curve  BMMf, 
the  extremities  M  and  M  of  the  ordi- 
nates  MP,  M'P',  will  describe  the  cir- 
cumferences of  two  circles,  and  the 
chord  MM'  will  describe  the  curved 
surface  of  the  frustum  of  a  cone.  The 
surface  of  this  frustum  is  equal  to 
(Geom :  Bk.  VIII,  Prop.  IV.) 

(circ.MP  +  circ.M'P') 


I 

M 
B 

M 

t>\ 

f 

P     Pf   X 


X  chord  MM' :         that  is,  to 


X  chord  MM' =  v  (MP  +MP')  X  chord  MM ; 


and  by  substituting  for  MP,  M'P'  their  values,  the  expres- 
sion for  the  area  becomes 

'  &  t  I7i +$£2+ &c-) chord  MM- 

If  now  we  pass  to  the  limit,  by  making  h  =  0,  the  chord 
MM  will  become  equal  to  the  arc  MM1  (Art.  128),  and  the 
surface  of  the  frustum  of  the  cone  will  coincide  with  that 
of  the  surface  described  by  the  curve  at  the  point  M.  If  we 
represent  the  surface  by  s  and  the  arc  of  the  curve  by  z, 
we  have,  after  passing  to  the  limit, 

ds  =  Zvydz, 
and  by  substituting  for  dz  its  value  (Art.  128),  we  have 


ds  —  2?ry  V  doc?  +  dyz : 

whence,  the  differential  of  a  surface  of  revolution  is  equal 
to  the  circumference  of  a  circle  perpendicular  to  the  axis, 
into  the  differential  of  the  arc  of  the  meridian  curve. 


INTEGRAL    CALCULUS.  263 

Remark.  It  should  be  observed  that  X  is  the  axis  about 
which  the  curve  is  revolved.  If  it  were  revolved  about 
the  axis  Y,  it  would  be  necessary  to  change  x  into  y  and 
y  into  07. 

295.  If  a  right  angled  triangle  CAB  be  revolved  about 
the  perpendicular  CA,  the  hypothenuse  CB  will  describe 
the  surface  of  a  right  cone.  If  we  represent  the  base  BA 
of  the  triangle  by  6,  the  altitude  CA  by  h,  and  suppose 
the  origin  of  co-ordinates  at  the  vertex  of  the  angle  C,  we 
shall  have 

x  :  y  :  :  h  :  b  :     hence 

b  b  , 

y  =  —x     and     dy  =  —dx. 
h  h 

Substituting  these  values  of  y  and  dy,  in  the  general  for- 
mula, we  have 

,.       bx  ,      /T0  .  10       bx2 


and  integrating  between  the  limits  x  =  0  and  x  =  h,  we 

obtain 

CB 


surface  of  the  cone  =  nb 

•          AV 

=  circ.AB  x 

296.  If  a  rectangle  ABCD  be  revolved  around  the  side 
AD,  we  can  readily  find  the  surface  of  the  right  cylinder 
which  will  be. described  by  the-  side  BC. 

Let  us  suppose  the  axis  AD  =  h,  and  AB  —  b :  the 
equation  of  the  iine  DC  will  be  y  =  b:  hence,  dy  =  Q. 
Substituting  these  values  in  the  general  expression  of  the 
differential  of  the  surface,  we  have 


264  ELEMENTS    OF   THE 

and  taking  the  integral  between  the  limits   x  =  0,   x  —  Jit 
we  have 

surface  =2^bh  =  circ.  AB  x  AD. 

297.  To  find  the  surface  of  a  sphere,  let  us  take  the 
equation  of  the  meridian  curve,  referred  to  the  centre  as 
an  origin  :  it  is 


and  by  differentiating,  we  have 

xdx  +  ydy  =  0  ; 

hence 

,  xdx          ,     j  2 

dy  =  --     and     ay2  = 

y 

Substituting  for  dy  its  value,  in  the  differential  of  the 
surface 


ds  =  2 Try  yda?  -f  dy2, 
we  obtain 


s  =    2^         da?  +      dx2=f2irRdx  =  2*Rx+  C. 

If  we  estimate  the  surface  from  the  plane  passing  through 
the  centre,  and  perpendicular  to  the  axis  of  X,  we  shall 
have 

s  =  0     for     x  =  0,     and  consequently     C  =  0. 

Now,  to  find  the  entire  surface  of  the  sphere,  we  must 
integrate  between  the  limits  x  —  +  j^  and  x  —  —  R,  and 
then  take  the  sum  of  the  integrals  without  reference  to 
their  algebraic  signs,  for  these  signs  only  indicate  the  po- 
sition of  the  parts  of  the  surface  with  respect  to  the  plane 
passing  through  the  centre  of  the  sphere. 


INTEGRAL    CALCULUS.  265 

Integrating  between  the  limits 

x  =  0     and    x  —  -f  R, 
we  find 

and  integrating  between  the  limits  x  =  0  and  x  =  —  R, 
there  results 

s=-2*R2; 
hence, 

surface  =  4w.R2=:27rjR  x  2R ; 

that  is,  equal  to  four  great  circles,  or  equal  to  the  curved 
surface  of  the  circumscribing  cylinder. 

298.  The  two  equal  integrals 

indicate  that  the  surface  is  symmetrical  with  respect  to  the 
plane  passing  through  the  centre. 

299.  To  find  the  surface  of  the  paraboloid  of  revolution, 
take  the  equation  of  the  meridian  curve 

y2  =  2px, 
which  being  differentiated,  gives 


=yy  and      = 
p  p2 


Substituting  this  value  of  dx  in  the  differential  of  the  sur 
face  it  reduces  to 


266  ELEMENTS    OF    THE 

But  we  have  found  (Art.  217) 


hence, 


and  if  we  estimate  the  surface  from  the  vertex  at  which 
point  y  =  0,   we  shall  have, 

0  =  —--  +  C,      whence,       C=  —  -~-t 
o  3 

and  integrating  between  the  limits 

y=0,     y  =  b, 
we  have 


300.  To  find  the  surface  of  an  ellipsoid  described  by 
revolving  an  ellipse  about  the  transverse  axis. 
The  equation  of  the  meridian  curve  is 


whence 

B2xdx  B        xdx 


Ay  A  -y/^2 — a?2 

substituting  the  square  of  this  value  in  the  differential  of 
the  surface  and  for  y  its  value 


-—-x/  A*      r2 

A  VA  -or 


we  have 


=  2  Tr-    dx 


INTEGRAL    CALCULUS.  267 


and  if  we  represent  the  part  without  the  sign  of  the  inte- 
gral by  D,  and  make 

A^B,  =  R\ 

we  shall  have 

s  —  Dfdx^K*  —  x2. 

But  the  integral  of  dx  ^/R2  —  x2  is  a  circular  segment 
of  which  the  abscissa  is  x,  the  radius  of  the  circle  being 
jR.  If,  then,  we  estimate  the  surface  of  the  ellipsoid  from 
the  plane  passing  through  the  centre,  and  also  estimate  the 
area  of  the  circular  segment  from  the  same  point,  any 
portion  of  the  surface  of  the  ellipsoid  will  be  equal  to  the 
corresponding  portion  of  the  circle  multiplied  by  the  con- 
stant D.  Hence,  if  we  integrate  the  expression 


between  the  limits  x  =  0  and  x  =  A,  and  designate 
by  D'  the  corresponding  portion  of  the  circle  whose 
radius  is  Rt  we  shall  have 

—  surface  ellipsoid  =  D  x  IX; 

hence,  surface  ellipsoid  =  2JD  x  I? . 

301.  To  find  the  surface  described  by  the  revolution  of 
the  cycloid  about  its  base. 

The  differential  equation  of  the  cycloid  is 

,  _        ydy 


268  ELEMENTS    OF    THE 

Substituting  this  value  of  dx  in  the  differential  equation 
of  the  surface,  it  becomes 


Applying  formula  (E),  Art.  243,  we  have 


hence, 

s  = 

If  we  estimate  the  surface  from  the  plane  passing  through 
the  centre,  we  have  C  =  0,  since  at  this  point  s  =  Q 
and  y  —  2r.  If  we  then  integrate  between  the  limits 
y  =  2r  and  y  —  0,  we  have 

1         t  32   ^     i. 

6'  =  — surface  = — r"71"7?  hence, 

<w  O 

64      , 

5  =  surtace  = TT  r% 

3 

that  is,  the  surface  described  by  the  cycloid,  when  it  is 
revolved  around  the  base,  is  equal  to  64  thirds  of  the 
generating  circle. 

The  minus  sign  should  appear  before  the  integral,  since 
the  surface  is  a  decreasing  function  of  the  variable  y 
(Art.  31). 


INTEGRAL    CALCULUS. 


269 


Of  the  Cubature  of  Solids  of  Revolution. 

302.  The  cubature  of  a  solid  is  the  expression  of  its 
volume  or  content. 

303.  Let  u  represent  the  volume  or 
solidity  generated  by  the  area  ABMP, 
when  revolved  around  the  axis  AX.  If 
we  make  AP  —  x.  PPf  =  h.  we  have 
M'P'  =  F(x+h).  Now,  the  solid  gene-      / 
rated  by  the  area  ABMM'P',  will  ex- 
ceed the  solid  described  by  ABMP,  by     ~4 P     P'  X 
t.he  solid  described  by  the  area  PMM'P'. 

The  solid  described  by  the  area  ABMP  is  a  function  of 
07.  and  the  solid  described  by  the  area  ABMM'P'  is  a  simi- 
lar function  of  (x  +  h).  If  we  designate  this  last  by  ur, 
we  have 

du       (Pu    W1        (Pu      h3          ,0 
*"dx    +c£?  1.2  +5?  1.2.3  H 

hence,  the  solid  described  by  PMM'P'  is 
du,      (Pu    h2        (Pu      h3 


Let  us  now  compare  the  cylinder  described  by  the  rectan 
gle  P'M  with  that  described  by  the  rectangle  PC.     The 
equation  of  the  curve  gives 


hence,  since  PP  —  h, 

cylinder  described  by  P'M—  *[F(x)]2h, 
cylinder  described  by  PC  -  T  [F(x  +  h)]2h; 


270  ELEMENTS    OF    THE 

and  the  ratio  of  the  cylinders  is 


the  limit  of  which,  when  h  =  0,  is  unity. 

But  the  solid  described  by  the  area  PMM'P'  is  less 
than  one  of  the  cylinders  and  greater  than  the  other, 
hence,  the  limit  of  A  the  ratio,  when  compared  with  either 
of  them,  is  unity.  Hence, 

du  ,      d*u   >  du       d*u     h 


the  limit  of  which,  when  h  =.  0,  is 

du 
dx 


whence 


and  finally 

du  =  ny2dx  ; 

the  differential  of  the  solidity  ny^dx  being  a  cylinder  whose 
base  is  ny2  and  altitude  dx. 

304.  Remark.  The  differential  of  a  solid,  generated  by 
revolving  a  curve  around  the  axis  of  Y,  is 


305.  Let  it  be  required  to  find  the  solidity  of  a  right 
cylinder  with  a  circular  base,  the  radius  of  the  base  Vain<j 


INTEGRAL    CALCULUS.  271 

r  and  the  altitude  h.    We  have  for  the  differential  of  the 
solidity 


and  since  y  =  r,  it  becomes 


and  taking  the  integral  between  the  limits  x  =0  and  x  =  \ 
we  have 


which  expresses  the  solidity. 

306.  To  find  the  solidity  of  a  right  cone  with  a  circular 
base,  let  us  represent  the  altitude  by  h  and  the  radius  of 
the  base  by  r,  and  let  us  also  suppose  the  origin  of  co-or- 
dinates at  the  vertex.  We  shall  then  have 

r  12^2 

y  =  ~hx          y     l?x' 

and  substituting,  the  differential  of  the  solidity  becomes 


and  by  taking  the  integral  between  the  limits  x  =  0  and 
x  =  7z,  we  obtain 


that  is,  the  area  of  the  base  into  one  third  of  the  altitude. 

307.  Let  it  be  required  to  find  the  solidity  of  a  prolate 
spheroid,  (An  :  Geom  :  Bk.  IX,  Art.  33). 
The  equation  of  a  meridian  section  is 


18 


272  ELEMENTS    OF   THE 

which  gives 

hence  the  differential  of  the  solidity  is 


B2 

du  =  TT—  -2  (A2  — 


and  by  integrating 


If  we  estimate  the  solidity  from  the  plane  passing  through 
the  centre,  we  have  for  x  —  0,  u  —  0,  and  consequently 
C  =  0  ;  and  taking  the  integral  between  the  limits  x  =  0 
and  x  =  A,  we  have 


1  2 

~  solidity  =  —  *B2  x  A  ; 


and  consequently 


2 
solidity  =      7r.fi2  X  2 A. 


But  TTjB2  expresses  the  area  of  a  circle  described  on  the 
conjugate  axis,  and  2A  is  the  transverse  axis  :  hence, 
the  solidity  is  equal  to  two-thirds  of  the  circumscribing 
cylinder. 

308.  If  an  ellipse  be  revolved  around  the  conjugate  axis, 
it  will  describe  an  oblate  spheroid,  and  the  differential  of 
he  solidity  would  be 

du  = 


INTEGRAL    CALCULUS.  273 

and  substituting  for  x2,  and  integrating,  we  should  find 

2 

solidity  =  —  *  A2  x  2B  : 
<j 

that  is,    two-thirds    of  the  circumscribing  cylinder. 

309.  If  we  compare  the  two  solids  together,  we  find 

oblate  spheroid  :  prolate  spheroid  :  :  A  :  B. 

310.  If  we  make  A  =  B,  we  obtain  the  solidity  of  the 
sphere,  which  is  equal  to  two-thirds  of  the  circumscribing 
cylinder,  or  equal  to 


311.  Let  it  be  required  to  find  the  solidity  of  a  para- 
boloid.    The  equation  of  a  meridian  section  is 


and  hence  the  differential  of  the  solidity  is 
d  u  =  2  npxdx  ;     hence 
u  =  irpx2  +  C  ; 


and  estimating  the  solidity  from  the  vertex,  and  taking  the 
integral  between  the  limits  x  =  0  and  x  =  h,  and  designa- 
ting by  b  the  ordinate  corresponding  to  the  abscissa  x  =  hj 
we  have 

?rb2x  —  ; 

that  is,  equal  to  half  the  cylinder  having  an  equal  base 
and  altitude. 

312.  Let  it  be  required,  as  a  last  example,  to  determine 


274 


ELEMENTS    OF    THE 


the  solidity  of  the  solid  generated  by  the  revolution  of  the 
cycloid  about  its  base. 

The  differential  equation  of  the  cycloid  is 


hence  we  have 


which,  being  integrated  by  formula  (E)  Art.  243,  and  then 
by  Art.  226,  we  find  the  solidity  equal  to  five-eighths  of 
the  circumscribing  cylinder. 


Of  Double  Integrals. 

313.  Let  us,  in  the  first  place,  consider  a  solid  limited 
by  the  three  co-ordinate  planes,  and  by  a  curved  surface 
which  is  intersected  by  the  co-ordinate  planes  in  the  curves 
CB,  BD,  DC. 

Through  any  point  of 
the  surface,  as  M,  pass 
two  planes  HQF  and 
EPG  respectively  paral- 
lel to  the  co-ordinate  planes 
ZX,  YZj  and  intersect- 
ing the  surface  in  the 
curves  HMF  and  EMG. 
The  co-ordinates  of  the 
point  M  are 

AP=x,PMf=y,MM'=z. 


INTEGRAL    CALCULUS.  275 

It  is  now  evident  that  the  solid  whose  base  on  the  co-ordi- 
nate plane  YX  is  the  rectangle  AQM'P,  may  be  extended 
indefinitely  in  the  direction  of  the  axis  of  X  without  chang- 
ing the  value  of  y,  or  indefinitely  in  the  direction  of  Y 
without  changing  x.  Hence,  x  and  y  may  be  regarded 
as  independent  variables. 

If,  for  example,  we  suppose  y  to  remain  constant,  and  x 
to  receive  an  increment  Pp  =  h,  the  solid  whose  base  is 
the  rectangle  AQM'P,  will  be  increased  by  the  solid 
whose  base  is  the  rectangle  M'm'pP  ;  and  if  we  suppose 
x  to  remain  constant,  and  y  to  receive  an  increment 
Qq  =  k,  the  first  solid  will  be  increased  by  the  solid  whose 
base  is  the  rectangle  Qqn'M'. 

But  if  we  suppose  x  and  y  to  receive  their  increments 
at  the  same  time,  the  new  solid  will  still  be  bounded  by 
the  parallel  planes  epg,  hqf,  and  will  differ  from  the  prim- 
itive solid  not  only  by  the  two  solids  before  named,  but 
also  by  the  solid  whose  base  is  the  rectangle  n'M'm'N'. 
This  last  solid  is  the  increment  of  the  solid  whose  base  is 
the  rectangle  M'Ppm',  when  we  suppose  y  to  vary;  or 
the  increment  of  the  solid  whose  base  is  the  rectangle 
Qqn'M',  when  we  suppose  x  to  vary. 

Let  us  represent  by  u  the  solid  whose  base  is  the  rect- 
angle A  QM'P  ;  u  will  then  be  a  function  of  x  and  y,  and 
the  difference  between  the  values  of  the  increments  of  ut 
under  the  supposition  that  x  and  y  vary  separately  ;  and 
under  the  supposition  that  they  vary  together,  will  be  equal 
to  the  solid  whose  base  is  the  rectangle  n'M'm'N'.  By 
taking  this  difference  (Art.  83)  we  have 


dxdy         2dx*dy 


276  ELEMENTS    OF   THE 

hence, 

solid  n'N'm'M ...M      d?u       I    d?u          1    d?u 

~  +~ 


and  passing  to  the  limit,  by  making  h  =  0  and  k—  0,  the 

second  member  becomes    -r-  7-. 

dxdy 

As  regards  the  first  member,  the  rectangle 
n'N'm'M  =  hxk, 

and  the  altitude  of  the  solid  becomes  equal  to  MM  =  z 
when  we  pass  to  the  limit :  hence 

ffiu 

dxdy  ~ 

314.  Although  the  differential  coefficient 

d?u 
dxdy  ~ 

has  been  determined  by  regarding  M  as  a  function  of  two 
variables,  we  can  nevertheless  return  to  the  function  u  by 
the  methods  which  have  been  explained  for  integrating  a 
function  of  a  single  variable. 
For  we  have 

d(^\ 

dr  u          \dx/ 

dxdy          dy 
hence 


and  integrating  under  the  supposition  that  x  remains  cwi- 


INTEGRAL    CALCULUS.  277 

slant,  and  y  varies,  we  have 


dx 
whence 


dx 

and  if  we  integrate  this  last  expression  under  the  supposi- 
tion of  x  being  the  variable,  and  make    / ' X'  dx  =  X, 

u  =  fdxfzdy  +  X  +  Y. 

It  is  plain  that  the  constant,  which  is  added  to  complete 
the  first  integral,  may  contain  x  in  any  manner  whatever; 
and  that  which  is  added  in  the  second  integral,  may  contain 
y :  the  first  will  disappear  when  we  differentiate  with 
respect  to  y,  and  the  second  when  we  differentiate  with 
respect  to  x. 

The  order  of  integration  is  not  material.  If  we  first 
integrate  with  respect  to  x,  we  can  write 


dzu 


dxdy 
and  by  integrating,  we  find 

-^=fzdx,     u=fdyfzdx: 

hence  we  may  write 

u  =  ffzdy  dx,     or     u  =  ffzdxdy, 

which  indicates  that  there  are  two  integrations  to  be  per- 
formed, one  with  respect  to  x,  and  the  other  with  respect 
to  y. 


273  ELEMENTS    OF    THE 

315.  If  we  consider  the  differentials  as  the  indefinitely 
small  increments  of  the  variables  on  which  they  depend, 
we  may  regard  the   prism  whose  base  is  the  rectangle 
nrNfm'Mf,  as  composed  of  an  indefinite  number  of  small 
prisms,  having  equal  bases,  and  a  common  altitude  dz. 
Each  one  of  these  prisms  will  be  expressed  by  dxdydz, 
and  we  shall  obtain  their  sum  by  integrating  with  respect 
to  z  between  the  limits     z  =  0     and     z  =  MM' ',     which 
will  give 

J  dxdydz  =  zdxdy. 

316.  It  is  plain  that  zdx  is  the  differential  of  the  area 
of  the  section  made  by  the  plane  HQF  parallel  to  the 
co-ordinate  plane  ZX ;  and  consequently 

f  zdx  =  area,  of  the  section  HQF. 

Hence,  (fzdx)dy  is  equal  to  the  elementary  solid  in- 
cluded between  the  parallel  planes  HQF,  hqf,  or 

f(fzdx)dy  =ff  zdxdy 

is  equal  to  the  solid  which  is  limited  by  the  surface  and 
the  three  co-ordinate  planes.  If  we  consider  a  section 
of  the  solid  parallel  to  the  co-ordinate  plane  YZ,  we  have 
fzdy  =  area  of  the  section  EPG,  and  ff  zdxdy  —  solidity 
of  the  solid. 

317.  Let  us  suppose,  as  a  first  example,  that 

1 

~ 

we  shall  then  have 


INTEGRAL    CALCULUS.  279 

Let  us  now  integrate  under  the  supposition  that  x  is  con- 
stant ;  we  then  have 


_  , 

+  y2      x  x 

in  which  X1  represents  an  arbitrary  function  of  x.  If  we 
now  make  JX'dx  —  X,  and  integrate  again  under  the 
supposition  that  a?  is  a  variable,  we  have 


x  x 

The  integral  of    — tang"1—     is  obtained  in  a  series  by 
x  x 

substituting  the  value  of  (Art.  228), 

\i       t/        "  "'          " 

***&»  =  „- 


and  since,  in  integrating  with  respect  to  a?,  we  must  add 
an  arbitrary  function  of  y,  which  we  will  represent  by  Y, 
we  shall  obtain 


dxdy  y      y         y3 

-  + 


We  shall  obtain  the  same  result  by  integrating  in  the  in- 
verse order,  viz.,  by  first  supposing  y  to  be  constant. 
Under  this  supposition 

dx          1  !  x       ,,, 


280  ELEMENTS    OP    THE 

then  integrating  with  respect  to  x, 


y  y 

But  by  observing  that  (Trig.  Art.  XVIII), 

00         K  V 

tang      =-tang     • 


we  shall  have,  after  the  second  integration,  and  the  addi 
tion  of  an  arbitrary  function  of  a?, 


and  as  we  can  include  the  term    —  logy     in  the  arbitrary 
function  Y,  this  result  may  be  placed  under  the  form 

_x  y 


which  is  the  same  as  the  result  before  obtained,  as  may  be 

shown  by  placing  for  tang"1-^-  its  value,  multiplying  each 

dii  °° 

term  by  —  ,  and  integrating. 

318.  When  we  consider 

ffzdxdy 

as  expressing  the  solidity  of  a  solid,  it  is  necessary  to  con- 
sider the  limits  between  which  each  integral  is  taken,  and 
these  limits  will  depend  on  the  nature  of  the  solid  whose 
cubature  is  to  be  determined.  Let  it  be  equired,  for  ex 


INTEGRAL    CALCULUS.  281 

ample,  to  find  the  solidity  of  a  sphere,  of  which  the  centre 
is  at  the  origin  of  co-ordinates.  Designating  the  radius 
by  R,  we  have 


and  consequently, 

ffzdxdy  =ffdxdy  VR*-a?-y2. 

If  now,  we  suppose  y  constant,  and  make   R2  —  y2  =  R'2, 
and  then  integrate  with  respect  to  #,  we  have 

-a?-y2=fdxVR/2-x2, 


and  integrating  this  last  expression,  first  by  formula  (B) 
Art.  239,  and  then  by  Art.  220,  we  have 


and  substituting  for  R'2  its  value,  we  obtain 


It  should  be  remarked,  that  fzdx  expresses  the  area  of 
a  section  of  the  sphere  parallel  to  the  co-ordinate  plane 
ZXt  for  any  ordinate  y  =  A  Q,  and  to  obtain  this  area  we 
must  integrate  between  the  limits  x  =  0  and  x  =  QF. 
But  since  the  point  F  is  in  the  co-ordinate  plane  YX, 
we  have  for  this  point  z  =  0,  and  the  equation  of  the  sur- 
face gives 


therefore,  for  every  value  of  y  the  integral  fzdx  must  be 
taken  between  the  limits  x  =  0  and  x  —  VR2  —  y*>     Inte- 


282  ELEMENTS    OF    THE 

grating  between  these  limits  we  have 


since,  sm"^!)^  — 


hence, 


and  taking  this  last  integral  between  the  limits  y  =  0  and 
y  =  A  C  =  R,  we  obtain 

irR* 
0     ' 

which  represents  that  part  of  tne  sphere  that  is  contained 
in  the  first  angle  of  the  co-ordinate  planes,  or  one-eighth 
of  the  entire  solidity.  Hence, 

4  1 

solidity  of  the  sphere  =  —  R*K  =  —  D37r. 

o  u 

We  might  at  once  find  the  solidity  of  the  hemisphere 
which  is  above  the  horizontal  plane  YX,  by  integrating 
between  the  limits 

a?  =  _  V.R2  -  y2    and    x  =  -f  VR2-y2. 
Taking  the  integral  between  the  limits 

x  =  0     and     x  =  —  V  R2  —  y2, 

we  have  fzdx  —  --  (R2  —  y2)  ; 

and  between  the  limits 

x  =  0     and     x  =  + 


INTEGRAL  CALCULUS.  283 

we  ha ve  Jzdx  =—(R2  —  y2)', 

hence,  between  the  extreme  limits,  we  have 


Then  taking  the  integral 


between  the  limits 

y  =  —  R     and    y  =  +  R, 
we  find  the  solidity  to  be 


or  the  solidity  of  the  entire  sphere  is, 


THE  END 


303 


